• 


UC-NRLF 

III  I Illl 

SB    35 


IN  MEMORIAM 
FLOR1AN  CAJORI 


PHILLIPS-LOOMIS  MATHEMATICAL    SERIES 


ELEMENTS    OF    TRIGONOMETRY 


PLANE    AND    SPHERICAL 


BY 

ANDREW    W.   PHILLIPS,    PH.D. 

AM> 

WENDELL    M.  STRONG,  PH.D. 

VALK    UN1VKRSITV 


NEW    YORK    AND    LONDON 

HARPER    &     BROTHERS     PUBLISHERS 
1899 


THE  PHILLIPS-LOOMIS  MATHEMATICAL  SERIES. 


ELEMENTS  OF  TRIGONOMETRY,  Plane  and  Spherical.  By 
ANDREW  W.  PHILLIPS,  Ph.D.,  and  WENDELL  M.  STRONG,  Ph.D.,  Yale 
University.  Crown  8vo,  Half  Leather. 

ELEMENTS  OF  GEOMETRY.  By  ANDREW  W.  PHILLIPS,  Ph.D., 
and  IRVING  FISHER,  Ph.D.,  Professors  in  Yale  University.  Crown 
8vo,  Half  Leather,  $1  75.  [By  mail,  $1  92.] 

ABRIDGED  GEOMETRY.  By  ANDREW  W.  PHILLIPS,  Ph.D.,  and 
IRVING  FISHER,  Ph.D.  Crown  8vo,  Half  Leather,  $1  25.  [By 
mail,  $1  40.] 

PLANE  GEOMETRY.  By  ANDREW  W.  PHILLIPS,  Ph.D.,  and  IRVING 
FISHEK,  Ph.D.  Crown  8vo,  Cloth,  80  cents.  [By  mail,  90  cents.] 

LOGARITHMIC  AND  TRIGONOMETRIC  TABLES.  Five- Place 
and  Four-Place.  By  ANDREW  W.  PHILLIPS,  Ph.D.,  and  WKNDKI.I. 
M.  STRONG,  Ph.D.,  Yale  University.  Crown  8vo. 

LOGARITHMS  OF  NUMBERS.  Five- Figure  Table  to  Accompany 
the  "Elements  of  Geometry,"  by  ANDREW  W.  PHILLIPS,  Ph.D.,  and 
IKVING  FISHER,  Ph.D.,  Professors  in  Yale  University.  Crown  8vo, 
Cloth,  30  cents.  [By  mail,  35  cents.] 


NEW    YORK    AND    LONDON  : 
HARPER    &    BROTHERS,   PUBLISHERS. 


Copyright,  1898,  by  HARPER  &  BROTHERS. 


All  rights  reserved. 


PREFACE 


IN  this  work  the  trigonometric  functions  are  defined  as 
ratios,  but  their  representation  by  lines  is  also  introduced  at 
the  beginning,  because  certain  parts  of  the  subject  can  be 
treated  more  simply  by  the  line  method,  or  by  a  combination 
of  the  two  methods,  than  by  the  ratio  method  alone. 

Attention  is  called  to  the  following  features  of  the  book : 

The  simplicity  and  directness  of  the  treatment  of  both 
the  Plane  and  Spherical  Trigonometry. 

The  emphasis  given  to  the  formulas  essential  to  the  solu- 
tion of  triangles. 

The  large  number  of  exercises. 

The  graphical  representation  of  the  trigonometric,  inverse 
trigonometric,  and  hyperbolic  functions. 

The  use  of  photo-engravings  of  models  in  the  Spherical 
Trigonometry. 

The  recognition  of  the  rigorous  ideas  of  modern  math- 
ematics in  dealing  with  the  fundamental  series  of  trigo- 
nometry. 

The  natural  treatment  of  the  complex  number  and  the 
hyperbolic  functions. 

The  graphical  solution  of  spherical  triangles. 

Our  grateful  acknowledgments  are  due  to  our  colleague, 
Professor  James  Pierpont,  for  valuable  suggestions  regard- 
ing the  construction  of  Chapter  VI. 

We  are  also  indebted  to  Dr.  George  T.  Sellew  for  making 
the  collection  of  miscellaneous  exercises. 

ANDREW  W.  PHILLIPS, 
WENDELL  M.  STRONG. 

YALE  UNIVERSITY,  December,  1808. 


TABLE    OF    CONTENTS 


PLANE   TRIGONOMETRY 
CHAPTER   I 

THE   TRIGONOMETRIC    FUNCTIONS 

PAGE 

Angles i 

Definitions  of  the  Trigonometric  Functions 4 

Signs  of  the  Trigonometric  Functions .8 

Relations  of  the  Functions 10 

Functions  of  an  Acute  Angle  of  a  Right  Triangle 13 

Functions  of  Complementary  Angles 14 

Functions  of  o°,  90°,  1 80°,  270°,  360° 15 

Functions  of  the  Supplement  of  an  Angle 16 

Functions  of  45°,  30°,  60° 17 

Functions  of  (  —  .r),  (180°— .r),  (i8o°+.r),  (360°—  .r) 18 

Functions  of  (90°—  y),  (90°  +j),  (270°—  y\  (270°+^) 20 

CHAPTER   II 

THE   RIGHT   TRIANGLE 

Solution  of  Right  Triangles 22 

Solution  of  Oblique  Triangles  by  the  Aid  of  Right  Triangles    .     .  28 

CHAPTER   III 

TRIGONOMETRIC   ANALYSIS 

Proof  of  Fundamental  Formulas  (i i)- (14) 32 

Tangent  of  the  Sum  and  Difference  of  Two  Angles 36 

Functions  of  Twice  an  Angle 36 

Functions  of  Half  an  Angle 36 

Formulas  for  the  Sums  and  Differences  of  Functions 37 

The  Inverse  Trigonometric  Functions 39 


vi  TABLE  OF  CONTENTS 

CHAPTER   IV 

THE  OBLIQUE  TRIANGLE 

PAGE 

Derivation  of  Formulas 41 

Formulas  for  the  Area  of  a  Triangle 44 

The  Ambiguous  Case 45 

The  Solution  of  a  Triangle  : 

(i.)  Given  a  Side  and  Two  Angles 46 

(2.)  Given  Two  Sides  and  the  Angle  Opposite  One  of  Them     .  46 

(3.)  Given  Two  Sides  and  the  Included  Angle 48 

(4.)  Given  the  Three  Sides 49 

Exercises 50 

CHAPTER  V 

CIRCULAR   MEASURE— GRAPHICAL   REPRESENTATION 

Circular  Measure 55 

Periodicity  of  the  Trigonometric  Functions 57 

Graphical  Representation 58 

CHAPTER   VI 

COMPUTATION  OF  LOGARITHMS  AND  OF  THE  TRIGONOMETRIC  FUNC- 
TIONS—DE   MOIVRE'S  THEOREM— HYPERBOLIC    FUNCTIONS 

Fundamental  Series 63 

Computation  of  Logarithms 64 

Computation  of  Trigonometric  Functions 68 

De  Moivre's  Theorem 70 

The  Roots  of  Unity 72 

The  Hyperbolic  Functions 73 

CHAPTER  VII 

MISCELLANEOUS   EXERCISES 

Relations  of  Functions    .     .  • 7§ 

Right  Triangles 80 

Isosceles  Triangles  and  Regular  Polygons 83 

Trigonometric  Identities  and  Equations 84 

Oblique  Triangles 88 


TABLE  OF  CONTENTS  vii 


SPHERICAL   TRIGONOMETRY 
CHAPTER  VIII 

RIGHT   AND   QUADRANTAL   TRIANGLES 

PAGE 

Derivation  of  Formulas  for  Right  Triangles 93 

Napier's  Rules 94 

Ambiguous  Case 97 

Quadrantal  Triangles 98 

CHAPTER   IX 

OBLIQUE-ANGLED   TRIANGLES 

Derivation  of  Formulas 100 

Formulas  for  Logarithmic  Computation 101 

The  Six  Cases  and  Examples 104 

Ambiguous  Cases 106 

Area  of  the  Spherical  Triangle 108 

CHAPTER   X 

APPLICATIONS  TO   THE  CELESTIAL  AND   TERRESTRIAL   SPHERES 

Astronomical  Problems no 

Geographical  Problems 113 

CHAPTER   XI 

GRAPHICAL   SOLUTION   OF   A   SPHERICAL   TRIANGLE 115 

CHAPTER   XII 

RECAPITULATION    OF   FORMULAS 119 

APPENDIX 

RELATION   OF  THE  PLANE,  SPHERICAL,  AND   PSEUDO-SPHERICAL 

TRIGONOMETRIES 125 


ANSWERS   TO    EXERCISES 129 


PLANE   TRIGONOMETRY 


CHAPTER   I 
THE  TRIGONOMETRIC    FUNCTIONS 

ANGLES 

1.  In  Trigonometry  the  size  of  an  angle  is  measured  by 
the  amount  one  side  of  the  angle  has  revolved  from  the 
position  of  the  other  side  to  reach  its  final  position. 

Thus,  if  the  hand  of  a  clock  makes  one-fourth  of  a  rev- 
olution, the  angle  through  which  it  turns  is  one  right  angle ; 
if  it  makes  one-half  a  revolution,  the  angle  is  two  right  an- 
gles; if  one  revolution,  the  angle  is  four  right  angles;  if  one 
and  one-half  revolutions,  the  angle  is  six  right  angles,  etc. 


O' 


B 

FIG.  2 


FIG.  3 


The  amount  the  side  OB  has  rotated  from  OA  to  reach  its  final  position 
may  or  may  not  be  equal  to  the  inclination  of  the  lines.  In  Fig.  i  it  is  equal 
to  this  inclination  ;  in  Fig.  4  it  is  not. 

Two  angles  may  have  the  same  sides  and  yet  be  different.     In  Fig.  2 

I 


PLANE    TRIGONOME  TR  1  ' 


and  Fig.  4  the  positions  of  the  sides  of  the  angles  are  the  same  ;  yet  in 
Fig.  2  the  angle  is  two  right  angles,  in  Fig.  4  it  is  six  right  angles.  The 
addition  of  any  number  of  complete  revolutions  to  an  angle  does  not  change 
the  position  of  its  sides. 

Question.  —  Through  how  many  right  angles  does  the  hour-hand 
of  a  clock  revolve  in  6^  hours?  the  minute-hand  ? 

Question.  —  If  the  fly-wheel  of  an  engine  makes  100  revolutions  per 
minute,  through  how  many  right  angles  does  it  revolve  in  i  second  ? 


Initial  line 


RIGHT   ANGLES 


Initial  line 


5}   RIGHT    ANGLES 


Def. — The  first  side  of  the  angle — that  is,  the  side  from 
which  the  revolution  is  measured — is  the  initial  line;  the 
second  side  is  the  terminal  line. 

Def. — If  the  direction  of  the  revolution  is  opposite  to  that 
of  the  hands  of  a  clock,  the  angle  is  positive;  if  the  same 
as  that  of  the  hands  of  a  clock,  the  angle  is  negative. 

Initial  line 


Initial  Line 

POSITIVE    ANGLE 


NEGATIVE   ANGLE 


The  angles  we  have  employed  as  illustrations—  those  described 

by  the  hands  of  a  clock—  are  all  negative  angles. 

2.  Angles  are  usually  measured  in  degrees,  minutes,  and 
seconds.  A  degree  is  one-ninetieth  of  a  right  angle,  a  min- 
ute is  one-sixtieth  of  a  degree,  a  second  is  one-sixtieth  of  a 
minute. 


THE    TRIGONOMETRIC   FUNCTIONS 


The  symbols  indicating  degrees,  minutes,  and  seconds  are  °  '  "; 
thus,  twenty-six  degrees,  forty-three  minutes,  and  ten  seconds  is 
written  26°  43'  10". 

3.  The  plane  about  the  vertex  of  an  angle  is  divided  into 
four  quadrants,  as  shown  in  the  figure;  the  first  quadrant 
begins  at  the  initial  line. 


in 


IV 


THE   FOUR  QUADRANTS 


II 


III 


ANGLE  IN  1ST  QUADRANT 


ANGLE  IN  2D  QUADRANT 


ANGLE  IN  3D  QUADRANT 


ANGLE  IN  4TH  QUADRANT 


An  angle  is  said  to  be  in  a  certain  quadrant  if  its  terminali 
line  is  in  that  quadrant. 

EXERCISES 

4.  (i.)  Express  2\  right  angles  in  degrees,  minutes,  and  seconds^ 
In  what  quadrant  is  the  angle? 

(2.)  What  angle  less  than  360°  has  the  same  initial  and  terminal 
lines  as  an  angle  of  745°? 

(3.)  What  positive  angles  less  than  720°  have  the  same  sides  as  am 
angle  of  —73°? 

(4.)  In  what  quadrant  is  an  angle  of  —890°? 


4  PLANE    TRIGONOMETRY 

DEFINITIONS    OF  THE    TRIGONOMETRIC   FUNCTIONS 

5.  The  trigonometric  functions  are  numbers,  and  are  de- 
fined as  the  ratios  of  lines. 

Let  the  angle  A  OP  be  so  placed  that  the  initial  line  is 
horizontal,  and  from  Py  any  point  of  the  terminal  line,  draw 
PS  perpendicular  to  the  initial  line. 


S    A 


ANGLE  IN  THE  1ST  QUADRANT 


ANGLE  IN  THE  2D  QUADRANT 


ANGLH  IN  THE  30  QUADRANT 


Denote  the  angle  A  OP  by  x. 
SP 


ANGLE   IN   THE  4TH  QUADRANT 


-      =  sine  of  x  (written  sin^r). 


=  cosine  of  x  (written  cos^r). 


THE    TRIGONOMETRIC  FUNCTIONS 


SP 

-y^  —  tangent  of  x  (written  tan  x). 

—  —  )=  cotangent  of  x  (written  cot  x). 
OP 


—  „ 
OP 


=  secant  of  x  (written  sec^r). 


=  cosecant  of  x  (written  csc^r). 

To  the  above  may  be  added  the  versed  sine  (written  versin)  and  coversed 
sine  (written  coversin),  which  are  defined  as  follows  : 

versiii  x  =  i  -  cos  a?;  coversiii  x  =  i  —  sin  x. 

The  values  of  the  sine,  cosine,  etc.,  do  not  depend  upon 
what  point  of  the  terminal  line  is  taken  as  P,  but  upon  the 
angle. 


S  S' 


S'S 


x 


For  the  triangles  OSP  and  OS'P'  being  similar,  the  ratio  of  any 
two  sides  of  OS'P'  is  equal  to  the  ratio  of  the  corresponding  sides 
of  OSP. 

Def. — The  sine,  cosine,  tangent,  cotangent,  secant,  and 
cosecant  of  an  angle  are  the  trigonometric  functions 
of  the  angle,  and  depend  for  their  value  on  the  angle 
alone. 

0.  A  line  may  by  its  length  and  direction  represent  a 
number;  the  magnitude  of  the  number  is  expressed  by  the 
Icngtli  of  the  line ;  the  number  is  positive  or  negative  ac- 
cording to  the  direction  of  the  line. 


PLANE   TRIGONOMETRY 


7.  In  §  5,  if  the  denominators  of  the  several  ratios  be 
taken  equal  to  unity,  the  trigonometric  functions  will  be  rep- 
resented by  lines. 

SP     SP 
Thus,  sin^r=  -~p=  -—  =  SP—  the  number  represented  by 

the  line,  that  is,  the  ratio  of  the  line  to  its  unit  of  length. 

Hence  SP  may  represent  the  sine  of  x. 

In  a  similar  manner  the  other  trigonometric  functions 
may  be  represented  by  lines. 

In  the  following  figures  a  circle  of  unit  radius  is  described 
about  the  vertex  O  of  the  angle  A  OP,  this  angle  being  de- 
noted by  x.  Then  from  §  5  it  follows  that 


FIG.  a 


Cot 


C       Cot         B 


FIG.  3 


FIG  4 


THE   TRIGONOMETRIC  FUNCTIONS  7 

SP  represents  the  sine  of  x. 
OS  represents  the  co§iiie  of  x. 
A  T  represents  the  tangent  of  x. 
BC  represents  the  cotangent  of  x. 
0/^represents  the  secant  of  x. 
OC  represents  the  co§ecant  of  x. 

For  the  sake  of  brevity,  the  lines  SP,  OS,  etc.,  of  the  preceding  figures  are 
often  spoken  of  as  the  sine,  cosine,  etc. 

Hence,  we  may  also  define  the  trigonometric  functions 
in  general  terms  as  follows : 

If  a  circle  of  unit  radius  is  described  about  the  vertex  of 
an  angle, 

(r.)  The  iine  of  the  angle  is  represented  by  the  perpendicular 
upon  the  initial  line  from  the  intersection  of  the  terminal  line  with 
the  circumference. 

(2.)  The  cosine  of  the  angle  is  represented  by  the  segment  of  the 
initial  line  extending  from  the  vertex  to  the  sine. 

(3.)  The  tangent  of  the  angle  is  represented  by  a  line  tangent  to 
the  circle  at  the  beginning  of  the  first  quadrant,  and  extending  from 
the  point  of  tangency  to  the  terminal  line. 

(4.)  The  cotangent  of  the  angle  is  represented  by  a  line  tangent 
to  the  circle  at  the  beginning  of  the  second  quadrant,  and  extending 
from  the  point  of  tangency  to  the  terminal  line. 

(5.)  The  secant  of  the  angle  is  represented  by  the  segment  of  the 
terminal  line  extending  from  the  vertex  to  the  tangent. 

(6.)  The  cosecant  of  the  angle  is  represented  by  the  segment  of 
the  terminal  line  extending  from  the  vertex  to  the  cotangent. 

The  definitions  in  §  5  are  called  the  ratio  definitions  of  the  trigonometric- 
functions,  and  those  in  §  7  the  line  definitions.  The  introduction  of  two 
definitions  for  the  same  thing  should  not  embarrass  the  student.  We  have 
shown  that  they  are  equivalent.  In  some  cases  it  is  convenient  to  use  the 
first  definition,  and  in  other  cases  the  second,  as  the  student  will  observe 
in  the  course  of  this  study.  It  is  therefore  important  that  he  should  be- 
come familiar  with  the  use  of  both. 


8 


PLANE   TRIGONOMETRY  ' 


SIGNS   OF   THE   TRIGONOMETRIC   FUNCTIONS 

8.  Lines  are  regarded  as  positive  or  negative  according 
to  their  directions.  Thus,  in  the  figures  of  §  5,  OS  is  posi- 
tive if  it  extends  to  the  rigJit  of  O  along  the  initial  line, 
negative  if  it  extends  to  the  left ;  SP  \s  positive  if  it  extends 
upward  from  OA,  negative  if  it  extends  downward.  OP,  the 
terminal  line,  is  always  positive. 

The  above  determines,  from  §  5,  the  signs  of  the  trigono- 
metric functions,  since  it  shows  the  signs  of  the  two  terms 
of  each  ratio. 

By  the  line  definitions  the  signs  may  be  determined  di- 
rectly. The  sine  and  tangent  are  positive  if  measured  up- 
ward from  OA,  and  negative  if  measured  downward. 

The  cosine  and  cotangent  are  positive  if  measured  to  the 
right  from  OB,  and  negative  if  measured  to  the  left. 

B         Cot  +  Cot-    B 


TIG.  3 


FIG.  4 


THE   TRIGONOMETRIC  FUNCTIONS 


The  secant  and  cosecant  are  positive  if  measured  in  the 
same  direction  as  the  terminal  line,  OP\  negative  if  measured 
in  the  opposite  direction. 

The  signs  of  the  functions  of  angles  in  the  different  quadrants  are  as  follows  : 


Quadrant 

I 

II 

Ill 

IV 

Sine  and  cosecant 

+ 
+ 

+ 

- 

- 

Cosine  and  secant 

- 

- 

+ 

Tangent  and  cotangent 

+ 

- 

+ 

- 

f>.  It  is  evident  that  the  values  of  the  functions  of  an 
angle  depend  only  upon  the  position  of  the  sides  of  the 
angle.  If  two  angles  differ  by  360°,  or  any  multiple  of  360°, 
the  position  of  the  sides  is  the  same,  hence  the  values  of 
the  functions  are  the  same. 


C    cot     B 


Thus  in  Fig.  i  the  angle  is  120°,  in  Fig.  2  the  angle  is  840°,  yet 
the  lines  which  represent  the  functions  are  the  same  for  both  angles. 

EXERCISE 

Determine,  by  drawing  the  necessary  figures,  the  sign  of  tan  1000°; 
cos  810°;  sin  760°;  cot  —70°;  cos  —  550°;  tan  —560°;  sec  300°;  cot 
1560°;  sin  130°;  cos  260°;  tan  310°. 


10 


PLANE   TRIGONOMETRY 


RELATIONS   OF  THE   FUNCTIONS 
10.  By  §  5,  whatever  may  be  the  length  of  OP,  we  have 


SP 


OS 


OP    smjr'  OP     °*x'  n^' 


SP 
— 


OS 


OP 


We  have,  then,  from  Figs.  2  and  3, 


SP  _  _  sinx 

OS-  - 


OS  COS  X 

-  —  f*Ol  JC  —  - 

SP~  " 


or 


Multiplying  (i)  by  (2), 

tana? 

i 

cot  x 

Again,  from  Figs.  2  and  3, 
OP 


tan  x 


_ 

co*x* 

OP  1 

—  =  C§C  X  =  —  -  . 

SP  sin  a? 

From  Figs.  2  and  3,  OS2-f  SP*=  OP\ 
or 
and 

Also,  O 
or  I  iKtan2o?=r  sec2.x; 


OP 


B       Cot 


sin'2jf  =  i  —  cos2jr  ;     cos9  x=  i  —  sin2  jr. 

*=OT\  and 


FIG.  3 


(0 

(2) 

(3) 


(4) 

(5) 

(6) 


(7) 
(8) 


THE   TRIGONOMETRIC  FUNCTIONS  u 

The  angle  x  has  been  taken  in  the  first  quadrant  ;  the 
results  are,  however,  true  for  any  angle.  The  proof  is  the 
same  for  angles  in  other  quadrants,  except  that  SP  be- 
comes negative  in  the  third  and  fourth  quadrants,  and  OS 
in  the  second  and  third. 

EXERCISES 

11.  (i.)  Prove  cos-r  sec.i-=  i. 
(2.)  Prove  sin^r  cscx  =  i. 
(3.)  Prove  tan  x  cos  x  —  sin  x. 

(4.)  Prove  sin  x  \f  \  —  cos'-  x  =  i  —  cos2.r. 

(5.)  Prove  tan  x  +  cot  .r  —  -  -  l-  -- 
sin.r  cos.r 

(6.)  Prove  sin4.r  —  Cos4.r  =  i  —  2  cos'.r. 

(7.)  Prove  —  =  sin.r. 

cot-r  sec.r 

(8.)  Prove  tan  x  sin  .r  -h  cosx  =  sec;tr. 

12.  The  formulas  (i)-(8)  of  §  10  are  algebraic  equations 
connecting  the  different  functions  of  the  same  angle.     If 
the  value  of  one  of  the  functions  of  an  angle  is  given,  we 
can  substitute  this  value  in  one  of  the  equations  and  solve 
to  find  another  of  the  functions.     Repeating  the  process,  we 
find  a  third  function,  etc. 

In  solving  equation  (6),  (7),  or  (8)  a  square  root  is  extracted; 
unless  something  is  given  which  determines  whether  to  choose  the 
positive  or  negative  square  root,  we  get  two  values  for  some  of 
the  functions.  The  reason  for  this  is  that  there  are  two  angles 
less  than  360°  for  which  a  function  has  a  given  value. 

EXERCISES 

13.  (i.)  Given  x  less  than  90°  and  sin.r  =  ^;  find  all  the  other 
functions  of  x. 

Solution.  — 


=r  rb  \/  i  —  1— 
Since  x  is  less  than  90°,  we  kno\v  that  cosx  is  positive. 


12  PLANE   TRIGONOMETRY 

Hence  cosx=  + 


$•3 

---- 

2 


(2.)  Given  tan.r  =  —  £  and  .r  in  quadrant  IV;  find  sin  x  and  cos  jr. 

Solution,  — 


_  _  , 
COS.T  ~ 
hence  3  sin  .*•=:  —  COSJF, 

sin2.*  +  cos2.*  =  i  ; 
hence  10  sin2jr  =  i  ; 


(3.)  Given  sin(  —  30°)  =  —  \\  find  the  other  functions  of  —  30°. 

(4.)  Given  x  in  quadrant  III  and  sin^r  =  —  £;  find  all  the  other 
functions  of  x. 

(5.)  Given  y  in  quadrant  IV  and  sin/=z  —  |,  find  all  the  other 
functions  of  y. 

(6.)  Given  cos  6o°  =  £;  find  all  the  other  functions  of  60°. 

(7.)  Given  sin  o°=:o;  find  coso°  and  tano°. 

(8.)  Given  tans'rrjand  z  in  quadrant  I;  find  the  other  functions 
of  JT. 

(9.)  Given  cot45°=  i  ;  find  all  the  other  functions  of  45°. 

(10.)  Given  tan/=£\/5  and  cos^>  negative;  find  all  the  other 
functions  of  y. 

(ii.)  Given  cot  30°=  \/3  i  fi°d  tne  other  functions  of  30°. 

(12.)  Given  2  sin-r=i  —  cos^r  and  x  in  quadrant  II;  find  sin^r 
and  cos  jr. 

(13.)  Given  tan.r4-cot;r  =  3  and  x  in  quadrant  I  ;  find  sin  jr.    i 


THE   TRIGONOMETRIC  FUNCTIONS  13 

FUNCTIONS   OF  AN    ACUTE   ANGLE   OF  A   RIGHT   TRIANGLE 

14.  The  functions  of  an  acute  angle  of  a  right  triangle 
can  be  expressed  as  ratios  of  the  sides  of  the  triangle. 


Remark.  —  Triangles  are  usually  lettered,  as  in  Fig.  2,  the  capital 
letters  denoting  the  angles,  the  corresponding  small  letters  the  sides 
opposite. 

In  the  right  triangle  ABC,  by  §  5, 

BC     a 


15.  From  §  14,  for  an  acute  angle  of  a  right  triangle,  we  have 

side  opposite  angle 
sme  =  --  s-s-  -  ^—  ; 
hypotenuse 

side  adjacent  to  angle 
cosine  =  -  —  ^  -  £—', 
hypotenuse 

tangent  =  .  sjde  opposite  angle 
side  adjacent  to  angle 

side  adjacent  to  angle 
cotangent  =  —  —,  —  -  —  '—  • 

side  opposite  angle 


PLANE   TRIGONOMETRY 


FUNCTIONS   OF  COMPLEMENTARY   ANGLES 

16*  From  §  14,  we  have 
sin  ^L  =  cos^ 


(o) 
tan  A  =  cot  B  =  cot  (9O°  -  A)  5 

cot  A  =  tan  B  =  tan  (9O°  —  A). 

Because  of  this  relation  the  sine  and  cosine  are  called  co-func- 
tions of  each  other,  and  the  tangent  and  cotangent  are  called  co- 
functions  of  each  other. 

The  results  of  this  article  may  be  stated  thus: 
A  function  of  an  acute  angle  is  equal  to  the  co-function  of 
its  complementary  angle. 

Ths  values  of  the  functions  of  the  different  angles  are  given  in  "  Trigo- 
nometric Tables."  By  the  use  of  the  principle  just  proved,  each  function 
of  an  angle  between  45°  and  90°  can  be  found  as  a  function  of  an  angle  less 
than  45°.  Consequently,  the  tables  need  to  be  constructed  for  angles  up  to 
45°  only.  The  tables  are  so  arranged  that  a  number  in  them  can  be  read 
either  as  a  function  of  an  angle  less  than  45°  or  as  the  co-function  of  the 
complement  of  this  angle. 

EXERCISES 

17.  (i.)  Express  as  functions  of  an  angle  less  than  45°: 

sin  70°  ;  cos  89°  30' ;          tan  63°  ; 

cos66°;          cot47°;  sin72°39'. 

(2.)  cosjr  =  sin  ix ;  find  x. 
(3.)  tan  x  =  cot  3-r ;  find  x. 
(4.)  sin  2^r  =  cos3-r ;  find  x. 
(5.)  cot(3O° — x)  =  tan(3o°-J- $x) ;  find  x. 
(6.)  A,  B,  and    C  are   the    angles  of  a  triangle;   prove  that 


Hint.  —  A+B  +  C=\  So0. 


THE    TRIGONOMETRIC  FUNCTIONS 


FUNCTIONS   OF  O°,  90°,  l8o°,  270°,  AND  360° 

18.  As  the   angle  x  decreases  towards  o°  (Fig.  i),  sin*  de- 
creases and  cos*  increases.     When  OP  comes  into  coincidence 
with  OA,  SP  becomes  o,  and  OS  becomes  OA(  =  i). 
Ik-nee  sino°  =  o.     coso°=i. 


FIG.  3 


FIG.  4 


As  the  angle  x  increases  towards  90°  (Fig.  2),  sin*  increases 
and  cos*  decreases.     When  OP  comes  into  coincidence  with  OB, 
SP  becomes  OB(—\)  and  OS  becomes  o. 
Hence  sinQO0^!,     cosgo^o. 

As  the  angle  x  decreases  towards  o°  (Fig.  3),  tan*  decreases 
and  cot*  increases.     When  OP  comes  into  coincidence  with  OA, 
^/"becomes  o  and  BC  has  increased  without  limit. 
Hence  tano°r=o,     coto°=:oo. 

As  the  angle  x  increases  towards  90°  (Fig.  4),  tan*  increases 
and  cot*  decreases.     When  OP  comes  into  coincidence  with  OB, 
AT  has  increased  without  limit,  and  BC—o. 
Hence 


Remark.  —  liy  coto°=co  we  mean  that  as  the  angle  approaches  indefinitely 
near  to  o°  its  cotangent  increases  so  as  to  become  greater  than  any  finite  quan- 
tity we  may  choose.  The  symbol  co  does  not  denote  a  definite  number,  but 
simply  that  the  number  is  indefinitely  great. 


i6 


PLANE    TRIGONOMETRY 


Tn  every  case  where  a  trigonometric  function  becomes  indefinitely 
great  it  is  in  a  positive  sense  if  the  angle  approaches  the  limiting 
value  from  one  side,  in  a  negative  sense  if  the  angle  approaches  the 
limiting  value  from  the  other  side.  Thus  cot  o°  = -|- oo  if  the  angle 
decreases  to  o°,  but  cot  o°=  —  oo  if  the  angle  increases  from  a  nega- 
tive angle  to  o°.  We  shall  not  often  need  to  distinguish  between 
H-oo  and  —oo,  and  shall  in  general  denote  either  by  the  symbol  oo. 

By  a  similar  method  the  functions  of  180°,  270°,  and  360°  may  be 
deduced.  The  results  of  this  article  are  shown  in  the  following  table  : 


Angle 

0° 

90° 

1  80° 

270° 

3600 

sin 

0 

I 

0 

—  I 

0 

cos 

i 

0 

-I 

O 

I 

tan 

o 

oo 

o 

00 

o 

cot 

00 

O 

CO 

0 

00 

19.  It  may  now  be  stated  that,  as  an  angle  varies,  its  sine  and  cosine 
can  take  on  "values  from  —  /  to  -\-  r  only,  its  tangent  and  cotangent  all 
values  from  —  oo  to  -\-  oo ,  its  secant  and  cosecant  all  values  from  —  oo 
to  -j-  oo ,  except  those  between  —  /  and  -f-  /. 


FUNCTIONS  OF  THE  SUPPLEMENT  OF  AN  ANGLE 
20.  Suppose  the  triangle  OPS  (Fig.  i)  equal  to  the  tri- 
angle OP'S '  (Fig.  2),  then  SP=S'Pf  and  OS=OS',  and  the 
angle  AOP'  (Fig.  2)  is  equal  to  the  supplement  of  A  OP 
(Fig.  i).  Also,  in  the  triangle  AOP'  (Fig.  3),  angle  AOP' 
=  angle  AOP'  (Fig.  2). 

V 


8'         O 

FIG.  2 


FIG.  3 


THE    TRIGONOMETRIC  FUNCTIONS 


It  follows  from  §§  5  and  8  that 

§in(l§O°—  x)  =  §in 

C0§  (1§O°—  05)  =  —  C  , 

tan  (1§0°  -  x)  =  -  tan  x ;    ' 
cot  (1§O°—  a?)  =  —  cot  x. 

The  results  of  this  article  may  be  stated  thus : 

The  sine  of  an  angle  is  equal  to  the  sine  of  its  supplement, 

and  the  cosine,  tangent,  and  cotangent  are  each  equal  to  minus 

the  same  functions  of  its  supplement. 

The  principle  just  proved  is  of  great  importance  in  the  solution  of  tri- 
angles which  contain  an  obtuse  angle. 

FUNCTIONS   OF  45°,    30°,   AND   60° 

21.  In  the  right  triangle  OSP  (Fig.  i)  angle  O  =  angle  P  =  4$°* 
and  OP— i. 

Hence  OS  =  SP  =  i  \/2. 

Therefore  sin  45°  =  00545°  =  £1/2;  §§14,16 

tan  45°  =  cot  45°=  i. 

P 


8 
FIG.   I 


c 

O 
FIG.  2 


In  equilateral  triangle  0/^4  (Fig.  2)  the  sides  are  of  unit  length 
PS  bisects  angle  OP  A,  is  perpendicular  to  OA,  and  bisects  OA. 
Hence,  in  the  right  triangle  OPS,  OS  =  %,  SP  =  ^\/^. 
Therefore  sin  30°  =  cos 60°  =  i;  §14 

cos  30°  =  sin  60°  =  i \/3  I 
tan  30°  =  cot  60°  =  J  y'3  ; 
cot  30°  =  tan  60°  =  \/3- 
2 


1 8  PLANE   TRIGONOMETRY 

22.  The  following  values  should  be  remembered  : 


Angle 

0° 

30° 

45° 

60° 

90° 

sin 

0 

i 

iv/2 

iVi 

I 

cos 

i 

4V5 

4^2 

k 

0 

EXERCISES 
Prove  that  if  x  =  30°, 

(i.)  sin  2x  =  2  sin^r  COS.T; 

(2.)  cos  3*  =  4  cos3  .r  —  3  cos  x  ; 

(3.)  cos  2;r  =r  cos'2  ;r —  sin2.r; 

(4.)  sin  3-r=  3  sin  x  cos2-r  —  sin3.r; 

2  tan  JT 

(5.)  tan2;r  = —          — . 
i  —  tan-jir 

(6.)  Prove  that  the  equations  of  exercises  i  and  3  are  cor- 
rect if  .r  =  450. 

(7)  Prove  that  the  equations  of  exercises  (2)  and  (4)  are  cor- 
rect if  .r=i200. 


The  following  three  articles,  §§  23-25,  are  inserted  for 
completeness.  They  include  the  functions  of  (90— ;r)  and 
(180— x\  which,  on  account  of  their  great  importance,  were 
treated  separately  in  §§  16  and  20. 


FUNCTIONS  OF  (—x),  (l8o°— x\  (l8o°  +  ;r),  (360° -*) 

23.  The  line  representing  any  function — as  sine,  cosine,  etc. 
— of  each  of  these  angles  has  the  same  length  as  the  line  repre- 
senting the  same  function  of  x. 

Thus  in  Figs.  2  and  3,  triangle  OS'P'  —  triangle  OSP,  hence  SP-^S'P', 


THE   TRIGONOMETRIC  FUNCTIONS 


FIG.   3 


FIG.  4 


In  Figs,  i  and  4,  triangle  OSP'  —  triangle  OSP,  hence  SP'=SF. 

In  Figs,  i,  2,  and  4,  triangle  OA  T  =  triangle  OA  T,  hence  A  T '  =  A  T. 

In  Figs,  i,  2,  and  4,  triangle  0/?C'  =  triangle  OBC.  hence  BC'=BC. 

Therefore  any  function  of  each  of  the  angles  (  —  x\  (180°—  x), 
(i8o°-f  •*),  (360°  —x\  is  equal  in  numerical  value  to  the  same  function 
of  x.  Its  sign,  however,  depends  on  the  direction  of  the  line  repre- 
senting it. 

Putting  in  the  correct  sign,  we  obtain  the  following  table: 


sin  (—  x]  —  —  sin  x 
cos  (—  x)  =  cos* 
tan(—  *)  =  —  tan* 
cot  ( —  x)  —  —  cot  x 

sin  (180°  +  *)=  —sin* 
cos  (180°  +  *)=  —  cos* 
tanCi8o°-f*)  =  tan* 
cot  (180°  +  *)  =  cot* 


sin  ( 1 80°  —  *)  =  sin  * 
cos  (180°  —  *)  —  —  cos* 
tan(i8o°  —  *)  —  —  tan* 
cot  (180°  -  *)  =  -  cot* 

sin  (360°  —  *)  =  —  sin  * 
cos  (360°  —  *)  —  cos* 
tan  (360°  —  *)  =  —  tan  * 
cot  (360°  —  *)  =  -  cot  * 


20 


PLANE   TRIGONOMETRY 


FUNCTIONS   OF  (9O° -7),  (9O°  +j),  (270°  -y\  (270°  +J/) 
£4.  The  line  representing  the  sine  of  each  of  these  angles  is 
of  the  same  length  as  the  line  representing  the'  cosine  of  y;  the 
cosine,  tangent,  or  cotangent,  respectively,  are  of  the  same  length 
as  the  sine,  cotangent,  and  tangent  of  y. 


FIG.  3 


For 


Triangle  OS'P'  =  triangle  OSP,  hence  S'P'  =  OS,  and  OS'  =  SP. 
Triangle  OA  T'  —  triangle  OBC,  hence  A  T'  =  BC. 
Triangle  OBC  —  triangle  OA  T,  hence  BC  —  AT. 

Therefore  any  function  of.  each  of  the  angles  (90°  —y\  (90°  -\-y\ 
(270°—^),  (270°+^),  is  equal  in  numerical  value  to  the  co-function 


THE   TRIGONOMETRIC  FUNCTIONS  21 

of  y.     Its  sign,  however,  depends  on  Ihe  direction  of  the  line  repre- 
senting it. 

Putting  in  the  correct  sign,  we  .obtain  the  following  table  : 
sin  (90°  —  y)  =  cos  v  sin  (90°  -f  y)  =  cosy 

cos (90°  —  y)  —  sin  r  cos (90°  +  y)  =  —  sinjv 

tan  (90°  —  r)  =  cot  v  tan  (90°  +}')=  —  cot  v 

cot  (90°  —y)  —-  tan  y  cot  (90°  +  y)  =  —  tan  y 

sin  (270°  —}')=  —  cosy  sin  (270°  +/)  =  —  cosy 

cos  (270°  —  y)  —  —  sin  y  cos  (270°  +^)  =  siny 

tan  (270°  — .r)  =  cot  y  tan  (270°  +y)  =  —  coty 

cot  (270°  -  y)  =  tan  r  cot  (270°  +  r)  =  —  tan  v 

25.  Either  of  the  two  preceding  articles  enables  us  directly  to- 
express  the  functions  of  any  angle,  positive  or  negative,  in  terms- 
of  the  functions  of  a  positive  angle  less  than  90°. 

Thus,  sin  21 2°  —  sin  (180°+  32°)=  —sin  32°; 

cos  260°  =  cos  (270°— 10°)  =  —  sin  10°. 

• 

EXERCISES 

(i.)  What  angles  less  than  360°  have  the  sine  equal  to  —  %\/2?  the- 
tangent  equal  to  \/3  ? 

(2.)  For  what  values  of  .r  less  than  720°  is  sin.r  =  .Jy^? 

(3.)  Find  the  sine  and  cosine  of  —30°;  765°;  120°;  210°. 

(4.)   Find  the  functions  of  405°;  600°;  1125°;  —45°;  225°. 

(5.)  Find  the  functions  of  —120°;  —225°;  —420°;  3270°. 

(6.)  Express  as  functions  of  an  angle  less  than  45°  the  functions  of 
233°:  -197°:  894°. 

(7.)  Express  as  functions  of  an  angle  between  45°  and  90°,  sin  267°;. 
tan  (  —  254°);  cos  950°. 

(8.)  Given  cos  164°  =  —  .96,  rind  sin  196°. 

(9.)  Simplify  cos (90° +  ,r) cos (2 70°  —  .r)  —  sin(i8o°  —  jr)sin(36o°  —  x).. 

(10.)  Simplifysin('8o°--'')tan(9o°  +  .r)+  .  .      '.         . 
y  sm  (270°  —  x}  sin2  (270°  —  x) 

(u.)  Express  the  functions  of  (.r — 90°)  in  terms  of  functions  of  x. 


CHAPTER    II 
THE   RIGHT   TRIANGLE 

27.  To  solve  a  triangle  is  to  find  the  parts  not  given. 

A  triangle  can  be  solved  if  three  parts,  at  least  one  of 
which  is  a  side,  are  given.  A  right  triangle  has  one  angle, 
the  right  angle,  always  given  ;  hence  a  right  triangle  can 
be  solved  if  two  sides,  or  one  side  and  an  acute  angle,  are 
also  given. 

The  parts  of  the  right  triangle  not  given  are  found  by 
the  use  of  the  following  formulas: 

opposite  side  adjacent  side 

(i)  sine        =-~  — ;       (2)  cosine        =—r  — ;      §  14 

hypotenuse  hypotenuse 


(3)  tangent^ 


opposite  side 


(4)  cotangent = 


_  adjacent  side  ^ 


16 


adjacent  side  '  opposite  side  ' 

To  solve,  select  a  formula  in  which  two  given  parts  enter;  substituting 
in  this  the  given  values,  a  third  part  is  found.  Continue  this  method  till 
all  the  parts  are  found. 

In  a  given  problem  there  are  several  ways  of  solving  the  triangle ;  choose 
the  shortest. 

EXAMPLE 

The  hypotenuse   of  a  right  triangle   is  47.653,  a  side  is 
21.34;  find  the  remaining  parts  and  the  area. 


THE   RIGHT   TRIANGLE 


SOLUTION    WITHOUT    LOGARITHMS 

The  functions  of  angles  are  given 


in  the  table  of 


Natural  Functions." 
21-34 


sin  A  =-= 

f    47.653 

47.653)21.3400^4478 
190612 

227880 
190612 
372680 
333571 
391090 
381224 
9866 

sin  A  =  .4478 

.4  =  26°  36' 
b—c  cos  A 
=47-  653  x  -8942 

47.653 
.8942 
95306 
190612 
428877 
381224 
42.6113126 
*  =42.61  f 

=  (90°  -26°  36'  1  =  63°  24 


.  34x42.61 


21-34 
42.61 

2134 
12804 


SOLUTION    EMPLOYING    LOGARITHMS 

It  is  usually  better  to  solve  triangles 
by  the  use  of  logarithms. 

The  logarithms  of  the  functions  are 
given  in  the  tables  of  "  Logarithms  of 
Functions."  * 


* 

sin  A  =  - 
c 


log  sin  A  =  log  a  —  log  c 
log  21.  34  =1.32919 

log  47.  653  =  1.67809 

---  sub. 
log  sin  .4=9.65110—  10 

A  =  26°  36'  14" 


cos  A=- 

log  b = log  c  +  log  cos  A 
log  47-  653  =  1-67809 
log  cos  26°  36'  14"  =9.95140—  10 
log  £=1.62949 

£=42.608 


J£=(9o°-26°  36'  I4")=63°  23'  46" 

area  =  \ab 
1  og  area = log  4  +  log  a  +  log  b 

log  ^  =  9.69897 -10 
Iog2i. 34=1. 32919 
log  42  608  =  i . 62949 
log  area=2. 65765 


2)909.2974 
454.6487 

area =4  54- 6 

*  In  this  solution  the  five-place  table  of  the  "  Logarithms  of  Functions"  is 
used. 

t  No  more  decimal  places  are  retained,  because  the  figures  in  them  are  not 
accurate  ;  this  is  due  to  the  fact  that  the  table  of  "  Natural  Functions"  is  only 
four- place. 


PLANE   TRIGONOMETRY 

CHECK  ON  THE  CORRECTNESS  OF  THE  WORK 


=  90.263  x  5.043 
90.263 
5-Q43 


270789 
361052 
45I3I5Q 
a*  =  455.  196309 

Extracting  the  square  root,  a  = 
21.34,  which  proves  the  solution  cor- 
rect. 


a  =  c-  -l>i  =  (c  +  b)(c  -  l>) 
=  90.261  x  5.045 

log  90.261  =  1.95550 

log  5.045  =  0.70286 

2)2.65836 

log  21. 34  =  1.32918 
a  =  21.34,  which  proves  the  solu- 
tion correct. 


Remark. — The  results  obtained  in  the  solution  of  the  preceding 
exercise  without  logarithms  are  less  accurate  than  those  obtained  in 
the  solution  by  the  use  of  logarithms ;  the  cause  of  this  is  that  four- 
place  tables  have  been  used  in  the  former  method,  five  place  in  the 
latter. 

EXERCISES 

28.  (i.)  In  a  right  triangle  £  =  96.42,  c=  114.81  ;  find  a  and  A. 

(2.)  The  hypotenuse  of  a  right  triangle  is  28.453,  a  side  is  18.197; 
find  the  remaining  parts. 

(3.)  Given  the  hypotenuse  of  a  right  triangle  =  747.24,  an  acute 
angle  =23°  45' ;  find  the  remaining  parts. 

(4.)  Given  a  side  of  a  right  triangle  =  37.234,  the  angle  opposite 
=  54°  27';  find  the  remaining  parts  and  the  area. 

. — .(5.)  Given  a  side  of  a  right  triangle  =  1.1293,  the  angle  adjacent 
=  74°  13'  27";  find  the  remaining  parts  and  the  area. 

(6.)  In  a  right  triangle  A  =  1 5°  22'  1 1 ",  c  —  .01 793  ;  find  b. 

(7.)  In  a  right  triangle  £  =  71°  34'  53",  £  =  896.33;  find  a. 

(8.)  In  a  right  triangle  c  =  3729.4,  £  =  2869.1  ;  find  A. 

(9.)  In  a  right  triangle  a  —  1247,  b—  1988  ;  find  c. 

(lo.)  In  a  right  triangle  (7  =  8.6432,  £  =  4.7815;  find  B. 

The  angle  of  elevation  or  depression  of  an  object  is  the 
angle  a  line  from  the  point  of  observation  to  the  object 
makes  with  the  horizontal. 


THE  RIGHT    TRIANGLE 


Thus  angle  x  (Fig.  i)  is  the  angle  of  elevation  of  P  if  O  is  the  point  of 

observation  ;  angle  y  (Fig.  2)  is  the  angle  of  depression  of  P  if  O  is  the 

point  of  observation. 

(n.)  At  a  horizontal  distance  of  253  ft.  from  the  base  of  a  tower  the 
angle  of  elevation  of  the  top  is  60°  20' ;  find  the  height  of  the  tower. 

(12.)  From  the  top  of  a  vertical  cliff  85  ft.  high  the  angle  of  depres- 
sion of  a  buoy  is  24°  31'  22";  find  the  distance  of  the  buoy  from  the 
foot  of  the  cliff. 

(13.)  A  vertical  pole  31  f t.  h igh  casts  a  horizontal  shadow  45  ft.  long ; 
find  the  angle  of  elevation  of  the  sun  above  the  horizon. 

(14.)  From  the  top  of  a  tower  115  ft.  high  the  angle  of  depression 
of  an  object  on  a  level  road  leading  away  from  the  tower  is  22°  13' 44"; 
find  the  distance  of  the  object  from  the  top  of  the  tower. 

(15.)  A  rope  324  ft.  long  is  attached  to  the  top  of  a  building,  and 
the  inclination  of  the  rope  to  the  horizontal,  when  taut,  is  observed 
to  be  47°  21'  17";  find  the  height  of  the  building. 

(16.)  A  light- house  is  150  ft.  high.  How  far  is  an  object  on  the 
surface  of  the  water  visible  from  the  top? 

[Take  the  radius  of  the  earth  as  3960  miles.] 

(17.)  Three  buoys  are  at  the  vertices  of  a  right  triangle;  one  side 
of  the  triangle  is  17,894  ft.,  the  angle  adjacent  to  it  is  57°  23'  46". 
Find  the  length  of  a  course  around  the  three  buoys. 

(i 8.)  The  angle  of  elevation  of  the  top  of  a  tower  observed  from  a 
point  at  a  horizontal  distance  of  897.3  ft.  from  the  base  is  10°  27'  42" ; 
find  the  height  of  the  tower. 

(19.)  A  ladder  42^  ft.  long  leans  against  the  side  of  a  building;  its 
foot  is  25!  ft.  from  the  building.  What  angle  does  it  make  with  the 
ground  ? 

(20.)  Two  buildings  are  on  opposite  sides  of  a  street  120  ft.  broad. 


S'.Z'b 


i 


26 


PLANE    TRIGONOMETRY 


The  height  of  the  first  is  55  ft. ;  the  angle  of  elevation  of  the  top  of 

the  second,  observed  from  the  edge  of  the  roof  of  the  first,  is  26°  37'. 

Find  the  height  of  the  second  building. 

A -(21.)  A  mark  on  a  flag-pole  is  known  to  be  53  ft.  7  in.  above  the 

j    ground.     This  mark  is  observed  from  a  certain  point,  and  its  angle. 

of  elevation  is  found  to  be  25°  34'.     The  angle  of  elevation  of  the  top 

of  the  pole  is  then  measured,  and  found  to  be  34°  17'.     Find  the 

height  of  the  pole. 

(22.)  The  equal  sides  of  an  isosceles  triangle  are  each  7  in.  long ;  the 

base  is  9  in.  long.     Find  the  angles  of  the  triangle. 


b  =  9 


Hint. — Draw  the  perpendicular  BD.  BD  bisects  the  base,  and  also  the 
angle  ABC. 

In  the  right  triangle  ABD,  AB—-]  in.,  AD—\\  in.,  hence  ABD  can 
be  solved. 

Angle  C=  angle  A,  angle  ABC— 2  angle  ABD. 

(23.)  Given  the  equal  sides  of  an  isosceles  triangle  each  13.44  in., 
and  the  equal  angles  are  each  63°  21'  42";  find  the  remaining  parts 
and  the  area. 

(24.)  The  equal  sides  of  an  isosceles  triangle  are  each  377.22  in., 
the  angle  between  them  is  19°  55'  32".  Find  the  base  and  the  area 
of  the  triangle. 

JL (25.)  If  a  chord  of  a  circle  is  1 8  ft.  long,  and  it  subtends  at  the  centre 

an  angle  of  45°  31'  10" ,  find  the  radius  of  the  circle. 

(26.)  The  base  of  a  wedge  is  3.92  in.,  and  its  sides  are  each  13.25  in. 
long;  find  the  angle  at  its  vertex. 


THE   RIGHT    TRIANGLE 


27 


(27.)  The  angle  between  the  legs  of  a  pair  of  dividers  is  64°  45',  the 
legs  are  5  in.  long;  find  the  distance  between  the  points. 

(28.)  A  field  is  in  the  form  of  an  isosceles  triangle,  the  base  of  the 
triangle  is  1793.2  ft. ;  the  angles  adjacent  to  the  base  are  each  53°  27' 
^49".  Find  the  area  of  the  field. 

6  (29.)  A  house  has  a  gable  roof.     The  width  of  the  house  is  30  ft., 
the  height  to  the  eaves  25!  ft.,  the  height  to  the  ridge-pole  33!  ft. 
Find  the  length  of  the  rafters  and  the  area  of  an  end  of  the  house. 
^     (30.)  The  length  of  one  side  of  a  regular  pentagon  is  29.25  in.  ;  find 
the  radius,  the  apothem,  and  the  area  of  the  pentagon. 


b 


Hint. — The  pentagon  is  divided  into  5  equal  isosceles  triangles  by  its  radii. 
Let  AOB  be  one  of  these  triangles.  ^#=29.25  in.;  angle  AOB=\  of 
36o°  =  72°.  Find,  by  the  methods  previously  given,  OA,  OD,  and  the  area 
of  the  triangle  A  OB. 

These  are  the  radius  of  the  pentagon,  the.  apothem  of  the  pentagon,  and 
\  the  area  of  the  pentagon  respectively. 

(31.)  The  apothem  of  a  regular  dodecagon  is  2  ;  find  the  perimeter. 
0(32.)  A  tower  is  octagonal ;  the  perimeter  of  the  octagon  is  153.7  ft. 
Find  the  area  of  the  base  of  the  tower. 

(33.)  A  fence  extends  about  a  field  which  is  in  the  form  of  a  regular 
polygon  of  7  sides;  the  radius  of  the  polygon  is  6283.4  ft.  Find  the 
length  of  the  fence. 

(34.)  The  length  of  a  side  of  a  regular  hexagon  inscribed  in  a  circle 
is  3.27  ft. ;  find  the  perimeter  of  a  regular  decagon  inscribed  in  the 
same  circle. 

(35.)  The  area  of  a  field  in  the  form  of  a  regular  polygon  of  9  sides 
is  483930  sq.  ft. ;  find  the  length  of  the  fence  about  it. 


28 


PLANE   TRIGONOMETRY 


SOLUTION   OF   OBLIQUE   TRIANGLES   BY   THE  AID   OF 

RIGHT  TRIANGLES 

29.  Oblique  triangles  can  always  be  solved  by  the  aid  of 
right  triangles  without  the  use  of  special  formulas ;  the 
method  is  frequently,  however,  quite  awkward  ;  hence,  in  a 
later  chapter,  formulas  are  deduced  which  render  the  solu- 
tion more  simple. 

The  following  exercises  illustrate  the  solution  by  means 
of  right  triangles : 

(i.)  In  an  oblique  triangle  ^  =  3.72,  ^  =  47°  52',  £'=109°  10';  find 
the  remaining  parts. 

The  given  parts  are  a  side  and  two  angles. 

C 


Hint.—  A  =  [i&o°-(B+C)], 

Draw  the  perpendicular  CD. 

Solve  the  right  triangle  BCD. 

Having  thus  found  CD,  solve  the  right  triangle  A  CD, 

(2.)  In  an  oblique  triangle  a  =  89.7,  c—  125.3,  B=  39°  8';  find  the 
remaining  parts. 

The  given  parts  are  two  sides  and  the  included  angle. 


125.3 


THE  RIGHT   TRIANGLE 


29 


'ii/.— Draw  the  perpendicular  CD. 
Solve  the  right  triangle  CBD. 
Having  thus  found  CD  and  AD(=c-DB),  solve  the  right  triangle  ACD. 

(3.)  In  an  oblique  triangle  a  =  3.67,  b  —  5.81,  A  =  27°  23';  find  the 
remaining  parts. 

The  given  parts  are  two  sides  and  an  anglt  opposite  one  of 
them. 

C 


B' 


B 


Either  of  the  triangles  ACB,  ACB'  contains  the  given  parts,  and 
is  a  solution. 

There  are  two  solutions  when  the  side  opposite  the  given  angle  is 
less  than  the  other  given  side  and  greater  than  the  perpendicular, 
CD,  from  the  extremity  of  that  side  to  the  base.* 

Hint.— Solve  the  right  triangle  ACD. 

Having  thus  found  CD,  solve  the  right  triangle  CDB  (or  CDB'\ 

(4.)  The  sides  of  an  oblique  triangle  are  0  =  34.2,  £  =  38 A  ^-  =  55. 12; 
find  the  angles. 

The  given  parts  are  the  three  sides. 


c  =55.12  0 

*  A  discussion  of  this  case  is  contained  in  a  later  chapter  on  the  solution 
of  oblique  triangles. 


Hint.— 
Hence 


PLANE    TRIGONOMETRY 


a*  -  ^  =  CF?  -  #•  -(c-  X}* 


In  each  of  the  right  triangles  A  CD  and  BCD  the  hypotenuse  and  a  side 
are  now  known ;  hence  these  triangles  can  be  solved. 

"jsi  (5-)  Two  trees,  A  and  B,  are  on  opposite  sides  of  a  pond.  The 
distance  of  A  from  a  point  C  is  297.6  ft.,  the  distance  of  B  from  C  is 
8644  ft.,  the  angle  ACS  is  87°  43'  12".  Find  the  distance  AB. 

(6.)  To  determine  the  distance  of  a  ship  A  from  a  point  B  on 
shore,  a  line,  EC,  800  ft.  long,  is  measured  on  shore  ;  the  angles,  ABC 
and  ACS,  are  found  to  be  67°  43'  and  74°  21'  16"  respectively.  What 
is  the  distance  of  the  ship  from  the  point  B? 

j^  (7.)  A  light-house  92  ft.  high  stands  on  top  of  a  hill;  the  distance 
from  its  base  to  a  point  at  the  water's  edge  is  297.25  ft.  ;  observed 
from  this  point  the  angle  of  elevation  of  the  top  is  46°  33'  15".  Find 
the  length  of  a  line  from  the  top  of  the  light-house  to  the  point. 

(8.)  The  sides  of  a  triangular  field  are  534  ft.,  679.47  ft.,  474.5  ft. 
What  are  the  angles  and  the  area  of  the  field  ? 

(9.)  A  certain  point  is  at  a  horizontal  distance  of  117^  ft.  from  a 
river,  and  is  u  ft.  above  the  river;  observed  from  this  point  the  angle 
of  depression  of  the  farther  bank  is  i°  12'.  What  is  the  width  of  the  river? 

(10.)  In  a  quadrilateral  ABCD,AB=  1.41,  BC—  1.05,  CD  =  1.76,  DA 
=  1.93,  angle  ^=75°  21';  find  the  other  angles  of  the  quadrilateral. 


\ 


n    Q  .  i  o 


•    THE  RIGHT   TRIANGLE  31 

Hint. — Draw  the  diagonal  DB. 

In  the  triangle  ABD  two  sides  and  an  included  angle  are  given,  hence  the 
triangle  can  be  solved. 

The  solution  of  triangle  ABD  gives  DB. 

I  laving  found  DB,  there  are  three  sides  of  the  triangle  DBC  known,  hence 
the  triangle  can  be  solved. 

(ii.)  In  a  quadrilateral  ABCD,  AB=\2.i,  AD  =  9.7,  angle  A  — 
47°  18',  angle  71  =  64°  49'»  angle  D=  100°;  find  the  remaining  sides 

Hint.— Solve  triangle  ABD  to  find  BD. 


*h 


CHAPTER   III 
TRIGONOMETRIC   ANALYSIS 

30.  In  this  chapter  we  shall  prove  the  following  funda- 
mental formulas,  and  shall  derive  other  important  formulas 
from  them  : 


§in  (x  +  y)  =  §in  x  co§  y  +  cos  x  §in  y, 
§in(a?-2/)  =  §in«  co§i/  -  cosx  siiii/, 

cos  (x  +  y]  =  cos  a?  cos?/  -sin  as  §iny, 

cos  (a?  —  y)  = 


(12) 
(13) 
(14) 


PROOF   OF   FORMULAS   (l  l)-(l4) 

31.  Let  angle  AOQ=  AT,  angle  QOP=y;  then  angle 


The  angles  *  and  j  are  each  acute  and  positive,  and  in  Fig.  i 
(•'*'+  y)  i§  less  tn^n  90°,  in  Fig.  2  (.r-f-j)  is  greater  than  90°. 


In   both  figures  the  circle  is  a  unit  circle,  and  SP  is  perpendicular  to 
OA ;  hence  SP=  sin  (x  +y),  OS=  cos  (x  +  y). 


TRIGONOMETRIC  ANALYSIS  33 

Draw  DP  perpendicular  to  OQ  ; 
then  DP=siny,     OD  =  cosy, 

angle  SPD  =  angle  AOQ  =  x. 

(Their  sides  being  perpendicular.) 

Draw  DE  perpendicular  to  OA,  DH  perpendicular  to  SP. 
Sin  (x  +/)  =  SP=  ED  +  PIP. 


cos/. 

(For  OED  being  a  right  triangle,  --  =  sin.r.) 


HP  '=  (cos  x)  x  DP—CQsx  sin/. 
HP 

(For  HPD  being  a  right  triangle,  -  =  cos  jr.) 


Therefore,  §in(a?  +  2/)  =  §in.r  co§?y  4-  cosx  siny.  (u) 

Cos  O  +/)  =  (95  =  OE  -  HD.  * 
=  (cos  x)  x  (9Z>  =  cos  x  cos/. 


(For  OED  being  a  right  triangle.  —  ~-  =  cos  jr.) 


(For  PHD  being  a  right  triangle,          =  sin  x.) 


Therefore,  cos  (x  +  y]  =  co§x  eos?/-§in^  §in//.  (13) 

5^.  The  preceding  formulas  have  been  proved  only  for 
the  case  when  x  and  y  are  each  acute  and  positive.  The 
proof  can,  however,  readily  be  extended  to  include  all  values 
of  x  and  y. 

Let/  be  acute,  and  let  x  be  an  angle  in  the  second  quad- 
lant  ;  then  x  =  (90°  4-  xr)  where  x'  is  acute. 
sin  (x  -f-/)  =  sin  (90°  +  x'  +y) 

=  cos(X+/)  §24 

=  cosx'  cos/  —  sin  x'  sin/ 

=  sin  (90°  +  x')  cos/  +  cos  (90°  4-  x')  sin  /      §  24 
•=smx  cos/4-  cos  x  sin/. 

*  If  (x  +,r)  is  greater  than  90°,  OS  is  negative. 


34  PLANE    TRIGONOMETRY 

Thus  the  formula  has  been  extended  to  the  case  where 
one  of  the  angles  is  obtuse  and  less  than  180°.  In  a 
similar  way  the  formula  for  cos(x+y)  is  extended  to  this 
case. 

By  continuing  this  method  both  formulas  are  proved  to 
be  true  for  all  positive  values  of  x  and  y. 

Any  negative  angle  y  is  equal  to  a  positive  angle  y'  ,  minus 
some  multiple  of  360°.  The  functions  of  y  are  equal  to 
those  of  y',  and  the  functions  of  (x-\-y)  are  equal  to  those 
of 


Therefore,  the  formulas  being  true  for  \x  -\-y'),  are  true  for 


A  repetition  of  this  reasoning  shows  that  the  formulas  are 
true  when  both  angles,  x  and  y,  are  negative. 

33.  Substituting  the  angle  —y  for  y  in  formula  (11),  it 
becomes 

s\n(x—y)  —  s'mx  cos(—  7)  +  cos;r  sin  (—y). 
But  cos(  —  y)  =  cosy,  and  sin(—  y)—  —  s\ny.      §23. 

Therefore,  sin  (a?  —  ?/)  =  §in.x  COST/  —  cosx  *m  //.  (12) 

Substituting  (—y)  for  y  in  formula  (13),  it  becomes 
cos  (x—y)  —  cos  x  cos  (  —y)  —  sin  x  sin  (—y), 


Therefore,  cos  (a?  -  y)  =  cos  x  cosy  +  sii»a?  siny.*  (14) 

EXERCISES 

34.  (i.)  Prove  geometrically  where  .r  and  j  are  acute  and  positive  : 

cos^y  —  cos^r  sinj/, 
n,r  sinj. 


*  Formulas  (12)  and  (14)  are  proved  geometrically  in  §  34.  The  geometric 
proof  is  complicated  by  the  fact  that  OD  and  DP  are  functions  of  —  y,  while 
the  functions  of  y  are  what  we  use. 


TRIGONOMETRIC  ANALYSIS 

.Q 

0,4— -x— ,H 


35 


Hint.—  Angle  AOQ-x,  angle  POQ=y,  and  angle  A  OP—  (x-y). 

Draw  /'/)  perpendicular  to  6>(). 

Then  DP=  sin  (—;•)  =  —sin  r  ;  but  Z>/*  is  negative,  therefore  PD  taken 
as  positive  is  equal  to  sin  y: 

OD=cof,(  —  ji')=cos  y, 

Angle  HTD  —  angle  AOQ=Jc.  their  sides  being  perpendicular. 
Draw  DI1  perpendicular  to  SP,  DE  perpendicular  to  OA. 

sin(x-y)=SP=£D-P//. 

From  right  triangle  OED,    £D.=(^'mx)x  OJr)=sinx  cos  jr. 
From  right  triangle  DHP,    P//=(cosx)x  PZ)=cosx  sin  y. 
Therefore,     .         sin  (.*•—;')=  sin  x  cos;/  —  cos  x  sin;  . 


From  right  triangle  0&D,      OE  =  (cos  x)x  0£>=cosx  cosjj'. 
From  right  triangle  DHP,     SJ//=(sin  x)  x  PD  =  ^\\\x  sin;'. 
Therefore,  cos(.r—  _j')=cos  x  cosj'  +  sin.r  sin;'. 

(2.)  Find  the  sine  and  cosine  of  (45°+,  r),  (30°—  x\  (6o°-f-,r),  in  terms- 
of  sin^r  and  cos.r. 

(3.)  Given  sin.r=$,  sin/  =  ^,  ,\-  and  y  acute;  find  sin(,r+j)  and 
sin(.r—  y). 

(4.)  Find  the  sine  and  cosine  of  75°  from  the  functions  of  30°  and  45°. 
Hint.—  75°=(45°  +  30°). 

(5.)  Find  the  sine  and  cosine  of  15°  from  the  functions  of  30°  and  45°. 

(6.)  Given  x  and  y,  each  in  the  second  quadrant,  sin  x  =  $,  siny  =  ^  ; 
find  sin  (x-\-y)  and  cos(.r  —  y}. 

(7.)  By  means  of  the  above  formulas  express  the  sine  and  cosine  of. 
(180°  —  .r),  (i8o°-|-,r),  (270°—  .r),  (270°+  •*),  in  terms  of  sin^r  and  cos-r, 

(8.)  Prove  sin  (6o°-f  45°)  +  cos  (60°  +  45°)  =  cos  45°. 

(9.)  Given  sin  45°  =  ^^/I,  cos  45°—  £  \/2  ;  find  sin  90°  and  cos  90°. 

(10.)  Prove  that  sin  (60°  -f-  x)  —  sin  (60°  —  .r)  = 


36  PLANE   TRIGONOMETRY 

TANGENT  OF  THE  SUM  AND   DIFFERENCE  OF  TWO  ANGLES 

_  sin(^r-f-j)      sin;r  cosj^-f-cos^r  sinj/ 
~~  cos(jr+7)~cos^  cosj  —  sin  x  si ny 
Dividing  each  term  of  both  numerator  and  denominator 
of  the  right-hand  side  of  this  equation  by  COSJT  cosj,  and 

remembering  that  ---  =  tan,  we  have 
cos 

tan  x  +  tan  y 


In  a  similar  way,  dividing  formula  (12)  by  formula  (14),  we 

.obtain 

tana?  -  tan?/ 
»>  =  !+ tan* 


FUNCTIONS   OF   TWICE  AN   ANGLE 
36.  An  important  special  case  of  formulas  (n),  (13),  and 

(15)  is  when  y~x\  we  then  obtain  the  functions  of  2x  in 

terms  of  the  functions  of  x. 

From  (n),  sin  (*?+.*)=  sin*  cos^-hcos^r  sin  jr. 
Hence  §in  2a?  =  2  §in  x  co§  x.  (17) 

From  (13),       co§2x  =  co§2a?-§in2x.  (18) 

Since        cos2^r=  i  —  sin8  JIT*  and  sinajr=  I  —  cos2.r, 

we  derive  from  equation  (18), 

cos  2^-=:  I—  2sin2^r,  (19) 

and  cos2;r  =  2  cos2^r—  I.  (20) 

From  (15),      tanto  =    >,lf.. 


FUNCTIONS   OF   HALF   AN  ANGLE 

57.  Equations  (19)  and  (20)  are  true  for  any  angle;  there- 
fore for  the  angle  \x. 

From(i9),  cos;r=  I  —  2  sin2^; 


TRIGONOMETRIC  ANALYSIS  37 

I  —  COS.T 


or 


therefore,  sin  £*  =  ±\~  •  (22) 

From  (20),  cos.r  =  2  cos2  4*  —  I  ; 

i  -f  cos^r 
or  cos  -£.*::=  ---  ; 

therefore,  cos-^^rfcy  —  *j~±-.  (23) 

Dividing  (22)  by  (23),  we  obtain 

(24) 


cos  a? 


FORMULAS   FOR   SUMS  AND    DIFFERENCES   OF   FUNCTIONS 

38.  From  formulas  (u)-(i4),  we  obtain 

sin  (x  +  /)-!-  sin  (x  —  j/)  =  2sin^r  cos}'  ; 
sin  (irH-^)—  .sin  (^  —  j/)  =  2cos;tr  sinj'  ; 
cos  (x  4-7)  +  cos  (*—}')  =  2  COS.T  cosj  ; 
cos  (x  +y)  —  cos  (x  —  y)  —  —  2sin^r  sinj/. 
Let  n  -  (x  +/)  and  v  —  (x  —y]  ; 

then  x  =  %(2i  +  ii),  y  —  %(u  —  v). 

Substituting  in  the  above  equations,  we  obtain 

sin  ?f  +  sinr  =  2  sin-|(«e  +v)cos^(u  —  v);  (25) 

siiifr  -  sin  r  =  2cos-|(/e  +  t')sin^(M  —  r);  (26) 

cos  M  +  c*o«  r=  2  co*lr(u+v)  cos|-(?«  —  f)  ;  (27) 

cos!/-cost'=-28iii-J(w  +  v)  §in-J(i«-f).          (28) 
Dividing  (25)  by  (26), 

§in  t  «  4-  sinv 


sin  M-  sin  v 


,     , 


EXERCISES 

,'i,9.  Express  in  terms  of  functions  of  x,  by  means  of  the  formulas 
of  this  chapter, 


38  PLANE   TRIGONOMETRY 

(i.)  Tan(i8o°  —  .r);  tan  (  1  80°  -f  x\ 

(2.)  The  functions  of  (x  —  180°). 

(3.)  Sin  (.r  —  90°)  and  cos  (^  —  90°). 

(4.)  Sin  (.r  —  270°),  and  cos  (x—  270°). 

(5.)  The  sine  and  cosine  of  (45°—  x);  of  (45°+.*). 

(6.)  Given  tan  45°=  i,  tan  30°  ^  £  -^3;  find  tan  75°;  tan  15°. 

cot./-  cot*/-l 

(7.)  Prove  cot  (05-1-  y)  —  -  -  —.  (30) 


Hint.  —  Divide  formula  (13)  by  formula  (n). 
COt.X  COt  91  +  1 

(8.)  Prove  cot  (x-y)  =  -  -  .  (31) 

cot  y-  cot  a? 

(9.)  Prove  cos  (30  +y)  —  cos  (30°  —y)  =  —  sin  y. 
(10.)  ProVe  sin  ^x  =  3  sin.  r  —  4  sin3.*-. 

/#»/.  —  Sin  3-r=sin  (x+2x). 
(n.)  Prove  cos  3*  =  4  cosfc—  3  cos  x. 
(12.)  If  x  and    y  are  acute    and    tan-r  =  £,  tanj/  =  J,  prove  that 


(i 3.)  Prove  that  tan  (.r-|~45°)  =  — 

i  —  tan  x 

(14.)  Given  siny=§  and  y  acute;  find  sin  \y,  cos^y,  and  tan  \y. 

(15.)  Given   cos^r=:— |  and  x  in  quadrant  II;    find    sin  2x  and 
cos  2.r. 

(16.)  Given   cos  45°  —  i  \/2  ;  find  the  functions  of  22!°. 

(17.)  Given  tan.r  =  2  and  .r  acute  ;  find  tan  \x. 

(i 8.)  Given  cos  30°  =  |  -^3  ;  find  the  functions  of  15°. 

(19.)  Given  cos9o°  =  o;  find  the  functions  of  45°. 
•*>  (20.)  Find  sin  §x  in  terms  of  sin  x. 

(21.)  Find  COS5-T  in  terms  of  COS.T. 

(22.)  Prove  sin(.r-j- y  -j-2-)  — sin  x  cosy  cos 5- -(-cos  ,r  sin  v  coss'-J-cos.r 
cosy  sin  z  —  sin  x  sin/  sin^. 

Hint. — Sin  (x+y  +  z)— sin  (x+y)  co$s  +  cos(.*+j')  sin  2. 

(23.)  Given  tan  2^  =  3  tan.r;  find  x. 
^    (24.)  Prove  sin  32° -|- sin  28°  =  cos  2°. 

(25.)  Prove  tan  x  -\-  cot  x  —  2  esc  2x. 

(26.)  Prove  (sin!.r-|-cos£..r)a=:  i  -)-sin^r. 

(27.)  Prove  (sin  \x  -  cos  l.r)-  =  i  -  sin  x. 


TRIGONOMETRIC  ANALYSIS  39 

^(28.)  Prove  cos  2.v  =  cos*.r  —  sin4.r. 
(29.)  Prove  tan  (45°  -f  x}  -f  tan  (45°  —  .r)  =  2  sec  2x. 

2  tan  -r 
<     (30.)  Prove  sin2.r  = — 


-f  tan2*' 

i  —  tan'.r 
(31.)  Prove  cos  2.1-  = 

i  -h  tan-. r 

I  4- sin  2x      /tan  .r-f-  i\' 
(32^  Prove  —  -  )  • 

i  —sin  2.x      Vtan.r—  i/ 

cin    r 

(33.)  Prove  tan|.r  = 


-|-  cos  .r 

sin  JT 
,    (34.)  Prove  coti.r=I-_c—. 

cos  x  —  cosy 

(35.)  Express  as  a  product  .* 

cos  x  -f-  cos_y 

COSJT  —  cos  r  _—  2  sin  •!(«*•+_;')  sin^(jr 

COS  a  +  COS/  2   COS  \  (x  •+•  1')  COS  ^  (x  — 

=  -tan^(j:+;')  tani(jr-.v). 

tan  ,r  -f  tan  y 

/  (36.)  Express  as  a  product  :    . 

cot  x  +  cot/ 

cos  (.1-  4-  y) 

(37.)  Prove   i  —  tan  x  tan  y  =:  -  — . 

'  - 


Till:   TXVKRSE   TRIGONOMETRIC   FUNCTIONS 

40.  Dcf. — The  expressions  sin— ^,  cos-^tan-'tf,  etc.,  de- 
note respectively  an  angle  whose  sine  is  a,  an  angle  whose 
cosine  is  a,  an  angle  whose  tangent  is  <7,  etc.  They  are 
called  the  inverse  sine  of  a,  the  inverse  cosine  of  <?,  the 
inverse  tangent  of  a,  etc.,  and  are  the  inverse  trigono- 
metric functions. 

Sin-'tf  is  an  angle  whose  sine  is  equal  to  a,  and  hence  de- 
notes, not  a  single  definite  angle,  but  each  and  every  angle 
whose  sine  is  a. 

*  Since  quantities  cannot  be  added  or  subtracted  by  the  ordinary  operations 
with  logarithms,  an  expression  must  be  reduced  to  a  form  in  which  no  addition 
or  subtraction  is  required,  to  be  convenient  for  logarithmic  computation. 


40  PLANE    TRIGONOMETRY 


Thus,  if  sin*=£,  jr=3o0,  150°,  (30°  +  360°),  etc., 

and  sin-  4=30°,  150°,  (3O°  +  36o°),  etc. 

Remark.  —  The  sine  or  cosine  of  an  angle  cannot  be  less  than  —  i 
or  greater  than  -|-  i;  hence  sin"1^  and  cos~'#  have  no  meaning  unless 
a  is  between  —  i  and  -f  i.  In  a  similar  manner  we  see  that  sec-'tf 
and  csc~la  have  no  meaning  if  a  is  between  —  i  and  -j-  *• 

EXERCISES 

41.  (i.)  Find  the  following  angles  in  degrees: 

sin~I|-v/2,  tan~T(—  *)»  sin~T(—  £). 

cos-1!,  cos-1!, 

(2.)  If  x  —  cot-^,  find  tan  x. 

(3.)  If  x  =  sin-xf  ,  find  cos  x  and  tan  x. 

(4.)  Find  sin  (tan-'i  \/3)- 
(5.)  Find  sin(cos-T£). 
(6.)  Find  cot  (tan-1  yS). 

(7.)  Given  sin-'fl  =  2  cos-1*?,  and  both  angles  acute  ;  find  a. 
(8.)  Given  sin"1^  =  cos-I^  ;  find  the  values  of  sin"1^  less  than  360°. 
(9.)  Given  tan~xi  =-}tan-fo,  and  both  angles  less  than  360°;    find 
the  angles. 

(lo.)  Given  sin"1^  —  cos-V?  and  sin-^  +  cos-1^  =  450°;  find  sin~V?. 
(n.)  Prove  sin  (cos~Vz)  =  ±  \/i—a*. 

Hint.  —  Let  jc=cos-1^  ;  then  a  =  cos  jc, 

sin  x=  ±  y  I  —  COS'JJT  =  ±  y  I  —  </2. 

(12.)  Prove  tan^an-^r+tan-1^)^: 

r.-b 
(13.)  Prove  tan(tan-rrt  —  tan-l^)=—  —  —  T- 

(14.)  Prove  cos(2  cos-I^)  =  2^2  —  i. 
(15.)  Prove  sin  (2  cos—1rt)  =  ±  2a  y/i  —  a". 

2(7 

(16.)  Prove  tan  (2  tan-1  a)=  ---  —• 

* 
(17.)  Prove  cos(2tan-I<7)= 


(i 8.)  Prove  sin(sin~Irt  +  cos-1^)  =  ab±.y/(\  —  a*}(\  — 


CHAPTER   IV 

THE    OBLIQUE    TRIANGLE 
DERIVATION   OF   FORMULAS 

42.  The  formulas  derived  in  this  and  the  succeeding 
articles  reduce  the  solution  of  the  oblique  triangle  to  its 
simplest  form. 

c  C  C 


FIG.  3 


Draw  the  perpendicular  CD.     Let  CD=/i, 

Then  -  —  sin  ^4; 

o 


and 


(In  Fig.  2    -=sin(i8o°-X)=--sin^) 

h 

-  —  sin  B. 


(In  Fig.  3  -= 


(32) 


By  division  we  obtain, 

a  _  sin  A 
b  ~  silTI* " 

Remark. — This  formula  expresses  the  fact  that  the  ratio  of  two  sides  of  an 
oblique  triangle  is  equal  to  the  ratio  of  the  sines  of  the  angles  opposite,  and 
does  not  in  any  respect  depend  upon  which  side  has  been  taken  as  the  base. 
Hence  if  the  letters  are  advanced  one  step,  as  shown  in  the  figure,  we  obtain, 
as  another  form  of  the  same  formula, 


42  PLANE   TRIGONOMETRY 

b  _  sin/? 

Repeating  the  process,  \ve  obtain 
c       sin  C 

;,=^A'  & 

The  same  procedure  may  be  applied  to  all  the  formulas  for  the  solution  of 
oblique  triangles.  Henceforth  only  one  expression  of  each  formula  will  lie  given. 

Formula  (32)  is  used  for  the  solution  of  triangles  in  which 
a  side  and  two  angles,  or  two  sides  and  an  angle,  opposite  one 
of  them  are  given. 

43.  We  obtain  from  formula  (32)  by  division  and  compo- 
sition, a  —  b  sin^— sin/? 


_ 

a  +  b  ~  sin  A  +  sin  B ' 

By  formula  (29),  denoting  the   angles   by  A    and   B,  in- 
stead of  u  and  z/, 

sin  A  -sin.#tan-J(^  —  B) 


Therefore,  ~~  -  ~-| -^  — '  ^  (33) 

This  formula  is  used  for  the  solution  of  triangles  in  which 
two  sides  and  the  included  angle  are  given. 

44.  Whether  A  is  acute  or  obtuse,  we  have  ^ 
C  C 


FIG. 


(If  A  is  acute  (Fig.  i),AD  =  bco*A,  DH  —  AB  -  AD  —  c  -  b  cos,-/,  CD  — 
bs\nA.     UA  is  obtuse  (Fig.  2),  AD  —  ^cos  (180°-^)  =  -  ^cos^,  DB—AB 
,  CD—  b  sin(i8o°—  A)-  b  sin^.) 


THE    OBLIQUE    TRIANGLE  43 


—  r  — 2  be  cosA+b*  (cos* A  +sinM). 

Therefore,  a2— 62+c2-  2bc  co§  A.  (34) 

This  formula  is  used  in  deriving  formula  (37). 

//  is  also  used  in  the  solution  without  logarithms  of  tri- 
angles of  which  two  sides  and  the  included  angle  or  three 
sides  are  given. 

45.  From  formula  (34),  cos^  =  —  — 7 — 
From  formula  (22),  §  37, 

2  sin24/2  =  i—  cos^4  =  I c  ~a  . 

2  be 

Hence          2  sin2 ±A  = 


2  be 

(b-_c 
~~2bc 


2bc 
Let  J==?±|.±f,  then  (a-6  +  c)  =  2(s-d),  and  (a  +  b-c) 


Substituting,  2  s\n*±A  =  - 


Hence  sin^^j  =  .A^)(S-  *\*  (35) 

V  be 

From  formula  (23),  §  37, 

2  COS'          = 


bc 

*  In  extracting  the  root  the  plus  sign   is  chosen  because  it   is  known  that 
sin  A  ,/  is  positive. 


44  PLANE    TRIGONOMETRY 

Hence  cos^A 

Dividing  (35)  by  (36),  we  obtain 

tan I  A  =.  \/(s~~ b^~ 

*  \/  \ 

s  {s  —  a) 


(36) 

(37) 


Let 


tan  i  ^  =  -  ~ 
s—a 


(38) 


Formulas  (37)  and  (38)  are  used  to  find  the  angles  of  a  tri- 
angle when  tJie  three  sides  are  given. 

FORMULAS   FOR   THE   AREA   OF   A   TRIANGLE 
40.   Denote  the  area  by  S. 
C 


D 

FIG.   I 


(In  Fig.  i,  CD=as,\\\B\  in  Fig.  2,  CD  -  « sin (180°-^)  =  asm£.) 

In  Figs.  I  and  2,         S=$c.CD. 

Hence  /S'^-JacsinB.  (39) 

From  formula  (17), 

sinZ?  =  2  sin    $  cosZ?. 


THE   OBLIQUE    TRIANGLE  45 

Substituting  for  sinj/?  and  cos^fi  the  values  found  in 
formulas  (35)  and  (36),  we  obtain 

sin£  =  —\  /s(s-a)(s-b}(s-e). 
ac* 

Therefore,  S=*/s(8  —  a)(s—b)(s— c).  (40) 

This  formula  may  also  be  written, 

S=sK.  (41) 

Formula  (39)  is  used  to  find  the  area  of  a  triangle  when 
two  sides  and  the  included  angle  are  knoivn;  formula  (40)  or 
formula  (41),  when  the  three  sides  are  known. 

THE   AMBIGUOUS    CASE 

47»  The  given  parts  are  two  sides,  and  the  angle  opposite 
one  of  them. 

Let  these  parts  be  denoted  by  a,  b,  A. 

C 


If  a  is  less  than  b  and  greater  than  the  perpendicular  CD 
(Fig.  i),  there  are  the  two  triangles  ACB  and  ACS',  which 
contain  the  given  parts,  or,  in  other  words,  there  are  two 
solutions. 

If  a  is  greater  than  b  (Fig.  2),  there  is  one  solution. 

If  a  is  equal  to  the  perpendicular  CD,  there  is  one  solu- 
tion, the  right  triangle  A  CD. 


46  PLANE    TRIGONOMETRY 

If  the  given  value  of  a  is  less  than  CD,  evidently  there 
can  be  no  triangle  containing  the  given  parts. 

Since  CD=t>sinA,  there  is  no  solution  when  «<  bs\\\A  ;  there  is  one 
solution,  the  right  triangle  A  CD  when  a=bs\\\A;  there  are  two  solutions 
when  a  <!  b  and  >  fisinA. 

48.  CASE  I. — Given  a  side  and  two  angles. 

EXAMPLE 
Given          a  =  36.738,  A  =  36°  55'  54",  B  =  72°  5'  56", 

C=i8o° — (A  +  £)=  180°—  109°  i'  50"  =  70°  58'  10". 

To  find  c. 

c sin  C 

a      sin  A 

=  1.56512 

log  sin  £=9.97559  — 10 
colog  sin  A  =0.22 1 23 
log  r=  i.  76194 
^•=57.80 


To  find  b. 


a      sin  /4 
log  rt  =  i.  56512 
log  sin  .#=9.97845  — 
colog  sin  A  =o.  221  23 
log  0=1.76480 
^  =  58.184 


Determine  b  from  r,  C,  and  B  by  the  formula 


This  check  is  long,  but  is  quite  certain  to  reveal  an  error.     A  check  which  is 
shorter,  but  less  sure,  is 

b  _  sin  B 

c      sin  C 

Solve  the  following  triangles  : 

(i.)  Given  a  —  567.25,  A  —  \\°  15',  ^  —  47°  12'. 

O  (2.)  Given  ^  =  783.29,  A  =  Si°  52',  ^  —  42°  27'. 

.    (3.)  Given  c=  1125.2,  A  =  79°  15',  ^=55°  n'. 

(4.)  Given  ^=15.346,  B=it°  51',  Cr=580  10'. 

(5.)  Given  a  =  5301.  5,  ^4  =69°  44',  C=4\°  18'. 

(6.)  Given  £=1002.1,  ^=48°  59',  €  =  76°  3'. 

t/f>.  CASE  II.  —  Given  two  sides  of  a  triangle  and  tlie  angle 
opposite  one  of  them. 


THE  OBLIQUE    TRIANGLE 


47 


EXAMPLE 
Given  a  =  23.203,  b  —  35.121,  A  =  36°  8'  10". 

C 


To  find  B  and  B  '. 


sin  A     a 
log  <$=i.  5.1556 
log  sin  ,4=9.77064—10 
colog  rz  =  8.  6344  5  —  10 
log  sin  #=9.95065  —  10 
£=63°  12' 


To  find  C  and'  C'. 
C  =  i8o°-(/f  +£)=So°  39'  50' 
')  =  2703-  50" 


To  find  c  and  c  . 
c     sin  C 


log  11=1.36555 
log  sin  (7=9.99421  —  10 
colog  sin  A  =0.22936 
log  f=i.  58912 
^=38.825 

log  a  =  i.  36555 
log  sin  C'  =9.65800  —  10 
colog  sin  A  =0.22936 
log  c'  =  i.  25291 
^'  =  17.902 


Check. 

Determine  b  from  c,  C,  and  B  by  the  formula 
b  —  a     tan4(/?- 


tan 


This  check  is  long,  but  is  quite  certain  to  reveal  an  error.     A  check  which  is 
shorter,  but  less  sure,  is 


c     sin  6" 

(i.)  How  many  solutions  are  there  in  each  of  the  following? 
(i.)  A  =  $o°,  a  =  i$,  6  =  20-, 
(2.)  A  =  30°,  a  =  TO,  £  =  20; 
(3.)  ^  =  30°,  a  =8,  6  =  20; 
(4.)  £  =  37°  23',  a  =  9.1,  6  =  7.$. 


v\ 


48 


PLANE    TRIGONOMETRY 


Solve  the  following  triangles,  finding  all  possible  solutions : 
7X2.)  Given  A  =  147°  12',  a  =  0.63735,  £  =  0.34312. 
(3.)  Given  A=  24°  31',  a  =  1.7424,  £  =  0.96245. 
(4.)  Given  A=  21°  21',  a  =  45.693,  £  =  56.723. 
(5.)Giveny?  =  61°  16',  # =  9.5124,  £  =  12.752. 
(6.)  Given  C=  22°  32',  a  =0.78727,  £  =  0.47311. 

£0.  CASE  III. — Given  two  sides  and  the  included  angle. 


Given  ^  =  41.003, 
parts  and  the  area. 


EXAMPLE 

'  =  48.718,  C  —  68°  33'  58";   find  the  remaining 


To  find  A  and  B. 

tan|(Z?  —  A)  _b  —  a 
~~ 


b-a  =    7.715 
b  +  rt  =  89.721 


log  (£-«)  =  0.88734 
colog  (b  +  a)  =  8.04710—10 
log  tan£(£  +  ^)  =  o.  16639 


log  tan  %(B  —  A  )  =  9.  10083  —  10 
-^^    7°  ii'  20" 


—  ^>2Q  54'  21" 
=48°  31'  41" 


r  _  sin  C 
a      sin  A 
loga=  1.61281 
log  sin  C=  9. 96888  -10 
colog  sin  A  =  o.  12535 
log<r=  1.70704 
c-    50.938 

To  find  the  area. 
S  =  \ab  sin  C 
=  9-69897  -10 
=  1.61281 
=  1.68769 
log  sin  C=  9. 96888  — 10 
log  S=  2.96835 
S=    929.72 


Check. 
sin  C       c 
siii~5  ~  7 
log  sin  B  =  9.94951  —  10 

log  c  =  1.70704 
coiog  b  —  8.31231  —  10 
log  sin  C  —  9.96886  —  10 


THE  OBLIQUE    TRIANGLE  49 

Solve  the  following  triangles,  and  also  find  their  areas  : 
/(7J  Given  A=  41°  15',  £=0.14726,  f =0.10971.   Q) 
— \^2.)  Given  C=  58°  47',  £=11.726,    #=16.147. 
(3.)  Given  £=  49°  50',  #  =  103  74,    ^=99.975. 
1  (4.)  Given  A=  33°  31',  £=0.32041,  ^=0.9203. 
(5.)  Given   C=i28°   7',  £=17.738,    #=60.571. 

51.  CASE  IV. — Given  the  three  sides. 

EXAMPLE 
Given  #  =  32.456,  £  =  41.724,  ^  =  53.987  ;  find  the  angles  and  area. 

^  =  64.084 
(.»•  —  </)  =  3i.628 
(s  —  ^  =  22.360 

log  A'=  i.  02349 


-r)-  10.097 


*-        /  (J  -  </JU  - 

A      \ 


-  u)(s  -  <>)(*  -  0 


log  (.f  —  rt)  =  1.50007 
lug  (-f  — ^)  =1.34947 

log  (s  —  r)=i. 00419 
colog  j  =  8. 19325  — 10 

2)2.0461*8 
log  A'=i.o2349 

To  find  A. 
K 


s — a 
log  A'=  i.  02  349 

log  (s-a)  =  1.50007 

sub. 

log  tan|^=g. 52342 -10 

A^  =  i8°  27'  23" 
^=36°  54  4"" 


log  (/-/;)  =  1.34947 

sub. 

log  tan  ^  .#=9.67402  -  i  o 
•§^=25°  16'  16" 
^=50°  32'  32" 


C* 


log  A'=  i.  02349 

log  (j-<-)=  1.00419 

sub. 

log  tan£C=o.oi93O 

|(7=46°  16'  22" 
C=92°  32'  44" 


Find  the  angles  and  areas  of  the  following  triangles: 

(i.)  Given  #  =  38.516,  £=44.873,  ^=14.517. 
„    (2.)  Given  #  =  2.1158,  £=3.5854,  ^=3.5679. 

*  C  could  be  found  from  (A  +J9)=(i8o°-  C),  but  for  the  sake  of  the  check  it 
is  worked  out  independently. 

4 


Check. 


50  PLANE    TRIGONOMETRY 

(3.)  Given  ^=82.818.  £=99.871,  ^=36.363. 

(4.)  Given  ^  =  36.789,  £=i  1.698,  ^=33.328. 

(5.)  Given  a  —  i  13.03,  £=131.17,  c  —  \  14.29. 

(6.)  Given  a=  .9763,  £=1.2489,  <•  =  1.6543. 

EXERCISES 

52.  (i.)  A  tree,  A,  is  observed  from  two  points,  B  and  C,  1863  ft. 
apart  on  a  straight  road.  The  angle  BCA  is  36°  43',  and  the  angle 
CBA  is  57°  21'.  Find  the  distance  of  the  tree  from  the  nearer 
point. 

(2.)  Two  houses,  A  and  B,  are  3876  yards  apart.  How  far  is  a  third 
house,  C,  from  A,  if  the  angles  ABC  and  J5AC  are  49°  17'  and  58°  18' 
respectively  ? 

(3.)  A  triangular  lot  has  one  side  285.4  ft.  long.  The  angles  adja- 
cent to  this  side  are  41°  22'  and  31°  19'.  Find  the  length  of  a  fence 
around  it,  and  its  area. 

(4.)  The  two  diagonals  of  a  parallelogram  are  8  and  10,  and  the 
angle  between  them  is  53°  8' ;  find  the  sides  of  the  parallelogram. 

(5.)  Two  mountains,  A  and  B,  are  9  and  13  miles  from  a  town,  C; 
the  angle  ACB  is  71°  36'  37".  Find  the  distance  between  the  moun- 
tains. 

(6.)  Two  buoys  are  2789  ft.  apart,  and  a  boat  is  4325  ft.  from  the 
nearer  buoy.  The  angle  between  the  lines  from  the  buoys  to  the 
boat  is  1 6°  13'.  How  far  is  the  boat  from  the  farther  buoy?  Are 
there  two  solutions  ? 

(7.)  Given  a  =  64.256,  r=  19.278,  C=i6°  19'  11";  find  the  differ- 
ence in  the  areas  of  the  two  triangles  which  have  these  parts. 

(8.)  A  prop  13  ft.  long  is  placed  6  ft.  from  the  base  of  an  embank- 
ment, and  reaches  8  ft.  up  its  face;  find  the  slope  of  the  embank- 
ment. 

(9.)  The  bounding  lines  of  a  township  form  a  triangle  of  which  the 
sides  are  8.943  miles,  7.2415  miles,  and  10.817  miles;  find  the  area 
of  the  township. 

(10.)  Prove  that  the  diameter  of  a  circle  circumscribed  about  a 
triangle  is  equal  to  any  side  of  the  triangle  divided  by  the  sine  of  the 
angle  opposite. 

'        -    i    -J    /       -    "'    Srf! 


'°    *~^t  b 

THE  OBLIQUE    TRIANGLE 


Hint. — By  Geometry,      angle  A  OB =2C. 

Draw  OD  perpendicular  to  AB. 
Angle  DOB=%AOB=C. 
DB=r  sin  DOB=r  sin  C. 
Hence  c=2rs'mC, 

c 
or  2r=~— ;. 

sine 

(ii.)  The  distances  AB,  BC,  and  AC,  between  three  cities,  A,  Bt 
and  Care  i$  miles,  14  miles,  and  17  miles  respectively.  Straight  rail- 
roads run  from  A  to  B  and  C.  What  angle  do  they  make  ? 

(12.)  A  balloon  is  directly  over  a  straight  road,  and  between  two 
points  on  the  road  from  which  it  is  observed.  The  points  are  15847 
ft.  apart,  and  the  angles  of  elevation  are  found  to  be  49°  12'  and 
53°  29'  respectively.  Find  the  distance  of  the  balloon  from  each  of 
the  points. 

(13.)  To  find  the  distance  from  a  point  A  to  a  point  B  on  the  op- 
posite side  of  a  river,  a  line,  AC,  and  the  angles  CAB  and  ACB  were 
measured  and  found  to  be  315.32  ft.,  58°  43',  and  57°  13'  respectively. 
Find  the  distance  AB. 

(14.)  A  building  50  ft.  high  is  situated  on  the  slope  of  a  hill.  From 
a  point  200  ft.  away  the  building  subtends  an  angle  of  12°  13'.  Find 
the  distance  from  this  point  to  the  top  of  the  building. 

(15.)  Prove  that  the  area  of  a  quadrilateral  is  equal  to  one-half 
the  product  of  the  diagonals  by  the  sine  of  the  angle  between 
them. 

(16.)  From  points  A  and  B,  at  the  bow  and  stern  of  a  ship  respec- 
tively, the  foremast,  C,  of  another  ship  is  observed.  The  points  A 
and  B  are  300  ft.  apart;  the  angles  ABC  and  BAC  are  found  to  be 

tftr) 

' 


52  PLANE    TRIGONOMETRY 

65°  31' and  1 10°  46' respectively.     What  is  the  distance  between  the 
points  A  and  C  of  the  two  ships  ? 

(17.)  Two  steamers  leave  the  same  port  at  the  same  time  ;  one  sails, 
directly  northwest,  12  miles  an  hour;  the  other  17  miles  an  hour,  in 
a  direction  67°  south  of  west.  How  far  apart  will  they  be  at  the  end 
of  three  hours  ? 

(i  8.)  Two  stakes,  A  and  />,  are  on  opposite  sides  of  a  stream;  a 
third  stake,  C,  is  set  92  ft.  from  A  ;  the  angles  ACB  and  CAB  are 
found  to  be  50°  3'  5"  and  61°  18'  20"  respectively.  How  long  is  a 
rope  connecting  A  and  Z?? 

(19.)  To  find  the  distance  between  two  inaccessible  mountain-tops, 
A  and  B,  of  practically  the  same  height,  two  points,  C  and  D,  are 
taken  one  mile  apart.  The  angle  CDA  is  found  to  be  88°  34',  the 
angle  DC  A  is  63°  8',  the  angle  CDB  is  64°  27',  the  angle  DCB  is  87°  9'. 
What  is  the  distance? 

(20.)  Two  islands,  B  and  C,  are  distant  5  and  V3  miles  respectively 
from  a  light-house,  A,  and  the  angle  BAC  is  33°"/>ff;  find  the  dis- 
tance between  the  islands. 

(21.)  Two  points,  A  and  B,  are  visible  from  a  third  point  C,  but 
not  from  each  other;  the  distances  AC,  BC,  and  the  angle  ACB  were 
measured,  and  found  to  be  1321  ft.,  1287  ft.,  and  61°  22'  respectively. 
Find  the  distance  AB.  \  ^  1  »^ 

(22.)  Of  three  mountains,  A,  B,  and  C,  B  is  directly  north  of  C  5 
miles,  A  is  8  miles  from  C  and  1 1  from  B.  How  far  is  A  south  of  B  ? 

(23.)  From  a  position  215.75  ft.  from  one  end  of  a  building  and 
198.25  ft.  from  the  other  end,  the  building  subtends  an  angle  of 
53°  37'  28";  find  its  length. 

(24.)  If  the  sides  of  a  triangle  are  372.15,  427.82,  and  404.17  ;  find 
the  cosine  of  the  smallest  angle. 

(25.)  From  a  point  3  miles  from  one  end  of  an  island  and  7  miles 
from  the  other  end,  the  island  subtends  an  angle  of  33°  55'  15''';  find 
the  length  of  the  island. 

(26.)  A  point  is  13581  in.  from  one  end  of  a  wall  12342  in.  long,  and 
10025  in-  from  tne  other  end.  What  angle  does  the  wall  subtend  at 
this  point? 

(27.)  A  straight  road  ascends  a  hill  a  distance  of  213.2  ft.,  and  is  in- 


THE  OBLIQUE   TRIANGLE  53 

clined  12°  2'  to  the  horizontal;  a  tree  at  the  bottom  of  the  hill 
subtends  at  the  top  an  angle  of  10°  5'  16".  Find  the  height  of  the 
tree. 

(28.)  Two  straight  roads  cross  at  an  angle  of  37°  50'  at  the  point  A  ; 
^  miles  distant  on  one  road  is  the  town  B,  and  5  miles  distant  on  the 
other  is  the  town  C.  How  far  are  B  and  C  apart  ? 

(29.)  Two  stations,  A  and  B,  on  opposite  sides  of  a  mountain,  are 
both  visible  from  a  third  station,  C\  AC  =11.5  miles,  BC  =  9.4  miles, 
and  the  angle  ACB—^cp  $1'.  Find  the  distance  from  A  to  B. 

(30.)  To  obtain  the  distance  of  a  battery,  A,  from  a  point,  B,  of  the 
enemy's  lines,  a  point,  C,  372.7  yards  distant  from  A  is  taken  ;  the  an- 
gles ACB  and  CAB  are  measured  and  found  to  be  j$  53'  and  74°  35' 
respectively.  What  is  the  distance  ABt 

(31.)  A  town,  B,  is  14  miles  due  west  of  another  town,  A.  A  third 
town,  C,  is  19  miles  from  A  and  17  miles  from  B.  How  far  is  C  west 
of  A? 

(32.)  Two  towns,  A  and  B,  are  on  opposite  sides  of  a  lake.  A  is 
18  miles  from  a  third  town,  C,  and  B  is  13  miles  from  C;  the  angle 
ACB  is  13°  17'.  Find  the  distance  between  the  towns  A  and  B. 

(33.)  At  a  point  in  a  level  plane  the  angle  of  elevation  of  the  top 
of  a  hill  is  39°  51',  and  at  a  point  in  the  same  direct  line  from  the  hill, 
but  217.2  feet  farther  away,  the  angle  of  elevation  is  2&J  53'.  Find 
the  height  of  the  hill  above  the  plane. 

(34.)  It  is  required  to  find  the  distance  between  two  inaccessi- 
ble points,  A  and  B.  Two  stations,  C  and  D,  2547  ft.  apart,  are 
chosen  and  the  angles  are  measured  ;  they  are  ACB=2j°  21',  BCD 
=33°  14',  £DA  =  i8°  17',  and  ADC=$i°  23'.  Find  the  distance  from. 
A  to  B. 

(35-)  Two  trains  leave  the  same  station  at  the  same  time  on  straight 
tracks  inclined  to  each  other  21°  12'.  If  their  average  speeds  are  40 
and  -^Smiles  an  hour,  how  far  apart  will  they  be  at  the  end  of  the  first 
fifteen  minutes? 

(36.)  A  ship,  A,  is  seen  from  a  light-house,  B\  to  determine  its  dis- 
tance a  point,  C,  300  ft.  from  the  light-house  is  taken  and  the  angles 
BCA  and  CBA  measured.  If  £CA  =  io8°  34'  and  CBA=6$Q  27',  what- 
is  the  distance  of  the  ship  from  the  light-house? 


54 


PLANE    TRIGONOMETRY 


(37.)  Prove  that  the  radius  of  the  inscribed  circle  of  a  triangle  is 
equal  to  a  sin-|Z>'  sin^Csec^. 


Hint.  —  Draw  OB,  OC,  and  the  perpendicular  OD. 
OB  and  OC  bisect  the  angles  B  and  C  respectively,  and  OD—r. 


sm 


Hence 


sin  \  /J  sin  -^  C      sin  ^  A'  sin  \  C 
sin  ^  v9  sin  • 


- 
cos-i/4 


=  a  sin 


sm  i  c  sec 


CHAPTER  V 

CIRCULAR  MEASURE— GRAPHICAL  REPRESENTATION 
CIRCULAR    MEASURE 

53.  The  length  of  the  semicircumference  of  a  circle  is 
irR  (77  =  3.14159  +  );  the  angle  the  semicircumference  sub- 
tends at  the  centre  of  the  circle  is  180°.  Hence  an  arc 
whose  length  is  equal  to  the  radius  will  subtend  the  angle 

1 80° 

— ;   this  angle   is  the   unit   angle   of   circular   measure, 

and  is  called  a  radian. 

7T  R 


If  the  radius  of  the  circle  is  unity,  an  arc  of  unit  length 
subtends  a  radian  ;  hence  in  the  unit  circle  the  length  of  an 
arc  represents  the  circular  measure  of  the  angle  it  subtends. 

Thus,  if  the  length  of  an  arc  is      ,  it  subtends  the  angle  -  radians. 


Since  one  radian  = 


1  80 


we  have 


00°  —      radians, 

2 

=  7r  radians, 


56  PLANE    TRIGONOMETRY 

270°=  -  radians, 

360°  — 2?r  radians/etc. 

The  value  of  a  radian  in  degrees  and  of  a  degree  in  radians  are  : 
i  radian  =  57.29578°, 

=  57°  17' 45". 
1°=:. 0174533  radian. 
In  the  use  of  the  circular  measure  it  is  customary  to  omit  the  word  radian  ; 

thus  we  write  -  ,  TT,  etc.,  denoting  -  radians,  TT  radians,  etc.      On   the  other 

hand,  the  symbols  °         are  always  printed  if  an  angle  is  measured  in  degrees, 
minutes,  and  seconds  ;  hence  there  is  no  confusion  between  tlie  systems. 

EXERCISES 

(i.)  Express  in  circular  measure  30°,  45°,  60°,  120°,  135°,  720°,  990°. 
(Take  ^=3.1416.) 

(2.)  Express  in  degrees,  minutes,  and  seconds  the  angles  -^,  —  ,  - ,-. 

8   10  2  4 

(3.)  What  is  the  circular  measure  of  the  angle  subtended  by  an  arc 
of  length  2.7  in.,  if  the  radius  of  the  circle  is  2  in.?  if  the  radius  is 

5  in.  ? 

<T4.  The  following  important  relations  exist  between  the 
circular  measure  x  of  an  angle  and  the  sine  and  tangent  of 
the  angle. 

(i .)  If  x  is  less  than  — ,  sin  x  <  x  <  tan  x. 


O  S 

Draw  a  circle  of  unit  radius. 
By  Geometry,        SP<arcAP<AT. 
Hence  sin  x  <x  <  tan^r. 


CIRCULAR  MEASURE  57 

sin  x          tan  x 
(2.)  As  x  approaches  the  limit  o,  —  —  and  —  —  approach 

&  Jv 

the  limit  i. 

Dividing  sin  x  <  x  <  tan  x  by  sin  x,  we  obtain 


, 
I  <-  -  < 


sinjr      cos^r 

sin  4:      cos  JIT 
Inverting,  i>~    —  > • 

*-v  1 

As  x  approaches  the  limit  o,  COS.T  approaches  the  length 
of  the  radius,  that  is,  i,  as  a  limit. 

Therefore,  -     -  approaches  the  limit  I. 

sin  x 
Dividing  i  >  —   -  >  cos^r  by  cos;r,  we  obtain 

Jv 

i          tan  x 


COS  X  X 

As  x  approaches  the  limit  o,  cos;r  approaches  the  limit  I ; 

hence  approaches  the  limit  I. 

cos-r 

Therefore,  -  approaches  the  limit  I. 


PERIODICITY   OF  THE   TRIGONOMETRIC   FUNCTIONS 

o&»  The  sine  of  an  angle  x  is  the  same  as  the  sine  of 
(^+360°),  (x  +  720°),  etc.— that  is,  of  (>+2;/7r),  where  n  is 
any  integer. 

The  sine  is  therefore  said  to  be  a  periodic*  function,  hav- 
ing the  period  360°,  or  2?r. 

The  same  is  true  of  the  cosine,  secant,  and  cosecant. 

*  If  a  function,  denoted  by /(.*)>  of  a  variable  jc,  is  such  that  f(x  +  k}=f(x} 
for  every  value  of  x,  k  being  a  constant,  the  function  f(x)  is  periodic;  if  k  is 
the  least  constant  which  possesses  this  property,  k  is  the  period  of  /(.r). 


58  PLANE    TRIGONOMETRY 

The  tangent  of  an  angle  x  is  the  same  as  the  tangent  of 
(x+  180°),  (^4-360°),  etc.— that  is,  of  (x  +  ntr\  where  n  is  any 
integer. 

The  tangent  is  therefore  a  periodic  function,  having  the 
period  1 80°,  or  TT. 

The  same  is  true  of  the  cotangent. 

GRAPHICAL    REPRESENTATION 

36.  On  the  line  OX  lay  off  the  distance  OA(=x)  to  rep- 
resent the  circular  measure  of  the  angle  x.  At  the  point  A 
erect  a  perpendicular  equal  to  sin  x.  If  perpendiculars  are 
thus  erected  for  each  value  of  x>  the  curve  passing  through 
their  extremities  is  called  the  sine  curve. 

If  sin*  is  negative,  the  perpendicular  is  drawn  downward. 


In  a  similar  manner  the  cosine,  tangent,  cotangent,  secant, 
and  cosecant  curves  can  be  constructed. 


Sine  Curve 


-1 


Cosine  Curve 


GRAPHICAL    REPRESENTA  TION 


59 


TangentCurve 


.0 


Cotangent  Curve 


6o 


PLANE    TRIGONOMETRY 


0 


3/27T 


SECANT   CURVE 


If  the  distances  on  OX  are  measured  from  O'  instead  of 
O,  we  obtain  from  the  secant  curve  the  cosecant  curve. 

In  the  construction  of  the  inverse  curves  the  number  is 
represented  by  the  distance  to  the  right  or  left  from  O\ 
the  circular  measure  of  the  angle  by  the  length  of  the  per- 
pendicular erected. 

All  of  the  preceding  curves,  except  the  tangent  and  co- 
tangent curves,  have  a  period  of  2?r  along  the  line  OX;  that 
is,  the  curve  extended  in  either  direction  is  of  the  same 
form  in  each  case  between  2?r  and  477%  4?r  and  6?r, — 27r  and 
o,  etc.,  as  between  o  and  2?r,  while  the  corresponding  inverse 
curves  repeat  along  the  vertical  line  in  the  same  period. 
The  period  of  the  tangent  and  cotangent  curves  is  TT. 


GRAPHICAL   REPRESENTATION 


61 


-i          o         +1 

INVERSE    SINE   CURVE 


-1 


INVERSE    COSINE   CURVE 


J L 


-3          -2  -1  0          +1          +  2          +3 

INVERSE    TANGENT    CURVE 


62 


PLANE    TRIGONOMETRY 


87T 


-3 


+  3 


INVERSE    SHCANT 


CHAPTER   VI 

COMPUTATION    OF    LOGARITHMS   AND    OF  THE   TRIG- 

ONOMETRIC FUNCTIONS  -DE   MOIVRE'S  THEOREM 

—HYPERBOLIC   FUNCTIONS 

*7f.  A  convenient  method  of  calculating  logarithms  and 
the  trigonometric  functions  is  to  use  infinite  series.  In 
work?  on  the  Differential  Calculus  it  is  shown  that 


=  x-~2+i-j-+  •  •  •  d) 

nf*$         OT^         ZM*^ 

=  x  -_+__-+...*  (2) 

/y»2  /y^          /y»6 

l-4jj  +  il-^!+       .  (3) 

Another  development  which  we  shall  use  later  is 
or      or^      or^      Gt*^ 

e*=:1  +  l!  +  2!  +  3!  +  4!+-  (4) 

where  c—  2.  7  18281  8  ...  is  the  base  of  the  Naperian  system 
of  logarithms. 

58.  The  series  (i)  converges  only  for  values  of  x  which  satisfy  the 
inequality  —  I<JT^I.  The  series  (2),  (3),  and  (4)  converge  for  all 
finite  values  of  .r. 

It  is  to  be  noted  that  the  logarithm  in  (i)  is  the  Naperian,  and  the 
angle  x  in  (2)  and  (3)  is  expressed  in  circular  measure. 

*  3!  denotes  1x2x3;  4]  denotes  1x2x3x4,  etc. 


64  PLANE    TRIGONOMETRY 

COMPUTATION    OF    LOGARITHMS 

59.  We  first  recall  from  Algebra  the  definition  and  some 
of  the  principal  theorems  of  logarithms. 

The  logarithm  to  the  base  a  of  the  number  m  is  the  number  .r 
\vhich  satisfies  the  equation, 


This  is  written  x  = 

The  logarithm  of  the  product  of  two  numbers  is  equal  to  the  sum 
of  the  logarithms  of  the  numbers. 

T  h  us  1  og^  m  11  =  \oga  m  +'1  oga  n. 

The  logarithm  of  the  quotient  of  two  numbers  is  equal  to  the  log- 
arithm of  the  dividend  minus  the  logarithm  of  the  divisor. 

in 

Thus  l°£<z —  —  1°R«;;/  —  l°gtf;z- 

n 

The  logarithm  of  the  power  of  a  number  is  equal  to  the  logarithm 
of  the  number  multiplied  by  the  exponent. 

Thus  log^  m*=p  \oga  m. 

To  obtain  the  logarithm  of  a  number  to  any  base  a  from  its  Na- 
perian  logarithm,  we  have 

log  in 

log*  m  = =  Ma  log,  m, 

tog,  a 

where  Mrt=  —   — ;  Ma  is  called  the  modulus  of  the  system. 

6*0.  We  proceed  now  to  the  computation  of  logarithms. 
The  series  (i)  enables  us  to  compute  directly  the  Naperian 
logarithms  of  positive  numbers  not  greater  than  2. 

Example.— To  compute  log*-  to  five  places  of  decimals. 
Substitute  -  for  x  in  (i): 


2/22     2          3     2'        4     2 

If  the  result  is  to  be  correct  to  five  places  of  decimals,  we  must  take  enough 
terms  so  that  the  remainder  shall  not  affect  the  fifth  decimal  place.     Now  we 


COMPUTATION  OF  LOGARITHMS 


know  by  Algebra  that  in  a  series  of  which  the  terms  are  each  less  in  numerical 
value  than  the  preceding,  and  are  also  alternately  positive  and  negative,  the  re- 
mainder is  less  in  numerical  value  than  its  first  term.  Hence  we  need  to  take 
enough  terms  to  know  that  the  first  term  neglected  would  not  affect  the  fifth 
place. 

Positive  terms 


2 

=0.5000000 

I 

1 

=  .0416667 

3 

2 

j 

I 

=  .0062500 

5 

2 

I 

1 

•=.  .OOIIl6l 

7 

21 

i 

1 

— 

=  .0002170 

9 

2 

i 

I 

=  .OOOO444 

1  1 

- 

i 

I 

=  .0000094 

13 

2 

.5493036 


Negative  terms 

i 

I 

2 

2*  ~~ 

3.1250000 

I 

4 

I 
'  21™ 

.0156250 

I 
6 

I 

.0026042 

i 
8 

I 

.0004883 

i 

T 

•  —  = 

.0000977 

10 

2 

I 

I 

12 

*2"  = 

.OOOO2O3 

I 

I 

14 

'2"= 

.0000044 

.1438399 


Subtracting  the  sum  of  the  negative  from  the  sum  of  the  positive  terms,  we 
obtain 


lo&^=  -4054637- 
Denote  the  sum  of  the  remaining  terms  of  the  series  by.  R.     Then,  by  Alge- 


bra, 


<  .0000021. 

The  error  caused  by  retaining  no  more  decimal  places  in  the  computation  is 
less  than  .0000006.  Hence  the  total  error  is  less  than  .0000027.  Therefore 
the  result  is  correct  to  five  decimal  places. 

61.  As  remarked,  the  series  (i)  does  not  enable  us  to 
calculate  directly  the  logarithms  of  numbers  greater  than  2, 
but  it  can  be  readily  transformed  into  a  series  which  gives 
us  the  logarithm  of  any  positive  number. 

Replacing  x  by  —  x  in  (i),  we  obtain 
5 


66  PLANE    TRIGONOMETRY 

x*      x3      x* 
log,  (i -*)=-*  --_-_- 

This  series  converges  for  —  i  <•*<*• 

Subtracting  this  from  (i),  we  obtain 


*v\ 

.,j-H  =2(^f+I+7V"-)'      (s) 

4  f  I 

/    which  converges  for  —  i  <  x  <  i . 

Putting  y=[-   -),  we  see  that  j  passes  from  o  to  oo  as  x 
\i     x/ 

passes  from  —  i  to  -f-i  ;   hence,  if  we  make  this  substitution  in 
(c),  we  get  a 


which  converges  for  all  positive  values  of  j,  and  therefore  enables 
us  to  compute  the  Naperian  logarithm  of  any  number. 

From   (5)  we   can   get   another  series  which  is   useful  :    put 

i  i4-x     y+i 

x—  -  ;  then,  as—  -=-=-  -  ,  equation  (5)  gives  us 


which  converges  for  all  positive  values  of  y.     Hence, 


This  series  gives  us  log,(jy-f-i),  when  log,^  is  known.  It  con- 
verges more  rapidly  than  (6),  when  y  is  greater  than  2,  and  hence 
should  be  used  under  these  circumstances. 

62.  To  construct  a  table  we  need  to  compute  directly 
only  the  logarithms  of  prime  numbers,  since  the  others  can 
be  obtained  by  the  relation 

log  ,ry  =  \og  ;r  +  log  y. 

i- 


i 


COMPUTATION  OF  LOGARITHMS 


67 


Thus,  to  obtain  the  logarithms  of  the  integers  up  to  10, 
we  need  to  compute  by  series  only  the  logarithms  of  the 
numbers  2,  3,  5,  and  7. 

(For  4=22,  6=2  .  3,  8  =  2*,  9=32,  10=2  .  5,  and  log  1=0.) 
In  this  case  we  are  computing  the  logarithms  of  successive  integers,  and 
should  therefore  use  (7). 

6Yf.  Example.  —  Compute  the  Naperian  logarithms  of  2,  3,  4,  and  5. 


.j 

3     3    33     5 


^.<..l+i.l9+.     \ 

3s     7    37     9     39  / 


-=-3333333 


.=.  0008230 
l-.  ^=.  0000653 

1.1^.0000056 
.3465729 

2 


Denote  the  sum  of  the  remaining 
terms  of  this  series  by  A'. 
Then,  by  Algebra, 


or        A"  <  .000000573. 

The  error  caused  by  not  retaining 
more  places  of  decimals  in  the  pre- 
ceding column  is  less  than  .0000005. 

Hence,  the  total  error  is  less  than 
.00000165. 


log,  2  =  .693  1458 

Remark. — We  should  get  the  same  series  if  we  were  to  use  (6). 


-  =  .2000000 
\  '  ^  =  .0026667 


6 
-  •  —=.0000018 

7  5' 


.2027325 

2 


.4054650 

Add  log,  2=  .6931458 
log,  3  =  1.0986108 


AX-  • 


or  A*  <  .00000006. 

Noting  the  errors  in  the  pre- 
ceding column  and  in  log,  2,  we 
see  that  the  total  error  is  less  than 

;OOOOO2I7. 


68  PLANE    TRIGONOMETRY 

Remark.  —  If  we  were  to  use  (6)  to  compute  log,  3,  we  should  have 


This  series  converges  much  more  slowly  than  the  above,  since  its 
terms  are  multiples  of  powers  of  \,  while  the  terms  of  the  above  are 
the  same  multiples  of  powers  of  \.  Thus,  we  should  be  obliged  to 
use  eight  instead  of  four  terms  to  have  the  result  correct  to  five 
places. 

log,  4  =  2  log,  2  =  1.3862916. 


or  log,  5  =  1.60944. 

64.  Proceeding  in  like  manner,  we  may  calculate  any  number  of 
logarithms. 

The  following  table  gives  the  Naperian  logarithms  of  the  first  ten 
integers  : 


log* J  —  -ooooo 
log,  2=  .69315 
log,  3  =  1.09861 
log*  4  =1.38629 
=  1.60944 


log,  6=  1.79176 

log,  7  =  I-94591 
log,  8  =  2.07944 
log,  9  =  2. 19722 
log,  10  =  2.30259 

The  common  logarithm  of  any  number  may  be  found  by  multiply- 
ing its  Naperian  logarithm  by  M10=. 43429448.  §  59 
Thus                 loglo  5  =  log,  5  X  43429448  =  .69897. 

65.  Remark. —  If  a  table  of  logarithms  were  to  be  computed,  the 
theory  of  interpolation  and  other  special  devices  would  be  employed. 

COMPUTATION   OF  TRIGONOMETRIC   FUNCTIONS 

c.  sin^r  cos^r 

f>6.  Since  tan;tr= ,  cot;r=  — ,  etc.,  the   computa- 

COS.T  sin  x 

tion  of  all  the  trigonometric  functions  depends  upon  that  of 
the  sine  and  cosine ;  thus  the  developments  (2)  and  (3)  suf- 
fice for  all  the  trigonometric  functions.  Further,  since  the 

^ 


COMPUTATION  OF   SINES  AND  COSINES  69 

sine  or  cosine  of  any  angle  is  a  sine  or  cosine  of  an  angle 

p-,  it  is  never  necessary  to  take  x  greater  than  -  in   the 
^4  4 

series  (2)  and  (3).  §  16 

Since  -  =0.785398  .,.<—,  these  series  converge  rapidly;  in  fact, 
4  10 

—  =  .000003   does   not   affect   the   fifth  decimal   place,  and   —  the 
9!  11! 

seventh. 

67.  Remark. — In  the  systematic  computation  of  tables  we  should 
not  calculate  the  functions  of  each  angle  from  the  series  independent- 
ly.    We  should  rather  make  use  of  the  formulas  (25)  and  (27)  of  §  38, 
thus  obtaining 

sinw.r  =  2  cos.r  sin  (n —  i)^r  —  sin  (n  —  2)  x, 
cos  nx  =  2  cos  x  cos  (n  —  i )  x  —  cos  (;/  —  2)  x. 

If  our  tables  are  to  be  at  intervals  of  i',  we  should  calculate  the 
sine  and  cosine  of  i'  by  the  series.  The  above  expressions  then  en- 
able us  to  find  successively  the  sine  and  cosine  of  2',  3',  4',  etc.,  till  we 
have  the  sine  and  cosine  of  all  angles  up  to  30°  at  intervals  of  i'. 

To  obtain  the  sine  and  cosine  of  angles  from  30°  to  45°  we  should 
make  use  of  these  results  by  means  of  the  formulas 
sin  (30°+  y)  =cosy  —  sin  (30°—  y), 
cos  (3o°-f-_y)  =  cos  (30°—^)  — sin  y. 

68.  To  employ  series  (2)  and  (3)  in  computing  the  sine 
and  cosine  we  must  first  convert  the  angle  into  circular 
measure. 

To  do  this  we  recall  that 

i°  =  . 017453293,     i '  =  .0002908882,     i  "  =  .  000004848 1 37. 
Example. — To  compute  the  sine  and  cosine  of  12°  15'  39". 

1 2°=  .209439516 
15'  =.004363323 
39"  =  .000189076 
12°  15'  39"  =  .213991915  in  circular  measure. 


PLANE   TRIGONOMETRY 


x-.  2139919 


( 


=  .  0000037 


.2139956 

X* 

subtract  —  =  .0016332 


Correct  to  five  decimal  places. 


cos*=i H — 

2!   4! 

1  =  1.0000000 


— =  .0000874 
4  ! 

1.0000874 
subtract  — -=  .0228963 

cos.r=  .9771911 
Correct  to  five  decimal  places. 


DE   MOIVRE'S    THEOREM 
00.   In  Algebra  we  learn  that  the  complex  number 


(8) 


may  be  represented  graphically  thus : 
Y 


Take  two  lines,  OX  and  OY,  at  right  angles  to  each  other. 
To  the  number  a  will  correspond  the  point  A,  whose  dis- 
tances from  the  two  lines  of  reference  are  ft  and  a  re- 
spectively. 

This  geometrical  representation  shows  at  once  that  we 
can  also  write  a  in  the  form 

a=r  (cos  $+4  sin  3).  (9) 

7O.  From  Algebra  we  recall  the  definition  of  the  sum  of  the 
complex  numbers  a  —  a  +  //3  and  b=y  +  il-,  namely 


Subtraction  is  defined  as  the  inverse  of  addition,  so  that 
a  —  b^a  —  y-M'()3  —  c). 


DE  MOIVRE'S   THEOREM  71 

Multiplication  is  most  conveniently  defined  when  a  and  b  are 
written  in  form  (9).     If 

a  —  r  (cos-$-M  sin  £)  and  d~s  (cos^-f  /  sin^), 
their  product  is  defined  by  the  equation 

ab  —  rs  [cos  (3  +  0)-|-/  sin  (£-r-0)]-     •  (IO) 

Division  is  defined  as  the  inverse  of  multiplication,  so  that 


Finally,  we  recall  that  in  an  equation  between  complex  numbers, 

•+173=7+13, 
we  have  «=y,    /3  =  S.  (n) 

*7  71.  Consider  the  different  powers  of  the  complex  number 

#  =  cos  $+*'  sin  $. 
By  (10)  we  have 

*a  =  (cos  $  +  /  sin  S)  (cos  $+/  sin  $), 

=  cos  2-S  +  /  sin  2-$. 
jc3=A:2  .  ^  =  (cos  2^+1  sin  2$)  (cos  $+*'  sin  3), 

=cos3^-|-/  sin  3^. 
And,  in  general,  for  any  integer  n, 

*«=:(cos  3+t  sin  ^)"=cos  n$+i  sin  n§. 

From  this  equation  we  have  De  Moivre's  Theorem,  which 
is  expressed  by  the  formula 

(12) 


72.  An  interesting  application  of  De  Moivre's  Theorem 
is  the  expansion  of  sin  nx  and  cos  nx  in  terms  of  sin  x  and 
COS.T.  Expanding  the  left-hand  side  of  (12)  by  the  bino- 
mial theorem,  and  substituting  x  for  3-,  we  have 

cosnx-\-i  sin  nx=cosnx+n  cos*"1  x  (i  sin  x)  -f  -  j  —  cosw~2^ 
(•  sin*)'  +  —  f*=!l£=!>  co8—«*(isin*)'  +  .  .  . 


72  PLANE    TRIGONOMETRY 

or 

cosnx  +  i  sinnx=(cosnx — j — -  cosw~2^  sin2*-}-  .  .  .) 

[//  (n—  i)  (n  —  2) 
n  cos""1  x  smx — cosw    3  x  sin  x+  .... 

Equating  real  and  imaginary  parts,  as  in  (11),  we  have 
cosnx=cosnx cosn~2x  sin'2#-f-  ;  .  .  (13) 


sin//*— ncosn     lxs'mx—  -  cos*"3*  sin3 x+.  . .  (14) 

Example. — n  =  5. 

cos  5-r  =  cos6 x  —  10  cos3.*-  sin2^r-f-5  cos^r  sin*,r. 
sin  5_r=  5  cos*,r  sin  or—  10  cos2  x  sin3  x  +  sin*  x. 

THE   ROOTS   OF   UNITY 

73.  We  find  another  application  of  De  Moivre's  Theorem 
in  obtaining  the  roots  of  unity.  The  #th  roots  of  unity  are 
by  definition  the  roots  of  the  equation 

xn—\. 

Every  equation  has  n  roots  and  no  more ;  hence,  if  we 
can  find  n  distinct  numbers  which  satisfy  this  equation  we 
shall  have  all  the  #th. roots  of  unity. 
Consider  the  //  numbers 

2?rr  2irr 

xr  =  cos \-t  sin , 

n  n 

r=o,  i,  2,  ...  n—  i. 

Geometrically  these  numbers  are  represented  by  the  n 
vertices  of  a  regular  polygon.  They  are,  therefore,  all  dif- 
ferent. We  shall  see  now  that  they  are  precisely  the  ;/th 
roots  of  unity. 

In  fact,  we  have  by  (12), 


*  —  ( cos  —  • — \-i  sin )  , 

\  ;/  n  ) 


THE   ROOTS   OF   UNITY 


(2Ttr\  .    .    .     /       2:rr\ 
n  .  -      )-f-j  sin  [«.—  -), 
«  /         V      «  / 


73 


sin  27T/-, 
=  1+*'.  0  =  1. 
Therefore  xr  is  one  of  the  roots  of  unity. 

Thus  the  cube  roots  of  unity  are  represented  by  the  points  A,  P, 
and  Q  of  the  following  figure.     In  the  figure  OA  =  i,  angle  AOP  = 

—  =  i2o°,  angle  AOQ  =  —  =  240°;  that  is,  the  circumference  is  di- 
vided into  three  equal  parts  by  the  points  A,  P,  and  Q.  Then  OD  =4, 
and  DP  —  DQ  =  ^^/^.  Hence  we  see  from  the  method  of  represent- 
ing a  complex  number  given  above  that  A  represents  -\-i,P  represents 
;,  Q  represents  —£  —  /'• 

P. 


EXERCISES 

74.  (i.)  Express  sin  4*  and  cos 4*  in  terms  of  sin  x  and  cos*. 
(2.)  Express  sin  6.r  and  cos  6*  in  terms  of  sin  x  and  cos*. 
(3.)  Find  the  six  6th  roots  of  unity. 

(4.)  Find  the  five  5th  roots  of  unity. 

THE   HYPERBOLIC   FUNCTIONS 

75.  The  hyperbolic  functions  are  defined  by  the  equations 

§inha?=— ~ — ,  (15) 


(16) 


cosh  x  =  — 


in  which  sinh*  and  cosh*  denote  the  hyperbolic  sine  and 


74  PLANE   TRIGONOMETRY 

hyperbolic  cosine  of  x  respectively.  These  functions  are 
called  the  hyperbolic  sine  and  cosine  on  account  of  their 
relation  to  the  hyperbola  analogous  to  the  relation  of  the 
sine  and  cosine  to  the  circle.  A  natural  and  convenient 
way  to  arrive  at  the  hyperbolic  functions  and  to  study  their 
properties  is  by  using  complex  numbers  in  the  following 
manner.  The  series  (2),  (3),  and  (4)  give  the  value  of  sin  x, 
cos^r,  and  e*  for  every  real  value  of  x.  These  series  also 
serve  to  define  sin^r,  cos^r,  and  ex  for  complex  values  of  x. 
In  the  more  advanced  parts  of  Algebra  it  is  shown  that 
the  following  fundamental  formulas  which  we  have  proved 
only  for  a  real  variable, 

sin  (x+y)  =  s\nx  cosjy-}-  cos  x  sin  y,  (17) 

cos  (x+y)  —  CQSx  cos^—  sin*  sinj,  (18) 

e*+*=e*e",  (19) 

hold  unchanged  when  the  variable  is  complex. 

This  fact  enables  us  to  calculate  with  ease  sin^r,  COS.T,  and 
ex  for  any  complex  value  of  the  variable. 

In  so  doing  we  are  led  directly  to  the  hyperbolic  func- 
tions. At  the  same  time  a  relation  between  the  trigono- 
metric and  hyperbolic  functions  is  established  by  means  of 
which  the  formulas  of  Chapter  III.  can  be  converted  into 
corresponding  formulas  for  the  hyperbolic  functions. 

Taking  x  and  y  real  and  replacing  y  in  (17),  (18),  and  (19)  by 

*>,  we  get 

sin  (aF+/y)  =  sina:  cos/y-f  cos*  sin  iy, 

cos  (x+iy)=cos  x  cos  iy—  sin  x  sin  iy, 


Thus  the  calculation  of  these  functions  when  the  variable 
is  complex  is  made  to  depend  upon  the  case  where  the  vari- 
able is  a  pure  imaginary. 


HYPERBOLIC  FUNCTIONS  75 

If  we  replace  x  by  ix  in  series  (4)  we  obtain 


V 


3!     5!     7! 

A  comparison  with  series  (2)  and  (3)  shows  that  these  two 
series  are  cos^r  and  sin^r  respectively;  hence  the  important 
formula  due  to  Euler — 

This  enables  us  to  calculate  ?*  from  sin^r  and  cos;r  when 
ix  is  a  pure  imaginary ;  that  is,  when  x  is  real. 

To  find  sin  ix  and  cosix  replace  x  in  (20)  by  ix\  we  obtain 

e—*=cosix+t  sin  ix.  (21) 

Again  replacing  x  by  —ix  in  (20),  we  obtain 

e* = cos  ix  —  i  sin  ix.  (22) 

The  sum  and  difference  of  (21)  and  (22)  give 

cos  ix  = ==  cosh  a?,  (23 ) 

(24) 


If  we  compute  the  value  of  e*  by  the  aid  of  series  (4)  for 
a  succession  of  values  of  x,  we  find  that  sinh^r  and  cosher 
are  represented  by  the  curves  on  page  76. 

The  system  of  formulas  belonging  to  the  hyperbolic  func- 
tions is  obtained  from  those  of  the  trigonometric  functions 
by  using  (23)  and  (24).  This  shows  that  for  every  formula 
in  analytic  trigonometry  there  exists  a  corresponding  for- 
mula in  hyperbolic  trigonometry  which  we  get  by  this  sub- 


76 


PLANE   TRIGONOMETRY 


stitution.  In  the  examples  which  follow,  this  method  is 
used  to  obtain  important  formulas  in  hyperbolic  trigonome- 
try. 

Replacing  x  by  —  ix  in  (23)  and  (24),  we  get 

-•  (25) 

-»  (26) 


which  are  formulas  frequently  used. 


Example.— sinh  (jr  -}-  j)  =  —  /  sin  / ( 

=  —  i  [sin  ix  cos  /p  +  cos  ix  sin  /y], 

=  —  /  [/'  sinh  .r  cosh^/  +  /  cosh  x  sinhj], 

=  sinh  x  cosh_y  +  cosh  x  sinh y. 

Example. — sinh  x -\-  sinh_y  =  —  /(sin  ix  -\-  sin  iy), 

—  —  i  2  sin  £  i(x  -\-y}  cos  ^  i  (x—  y\ 
=  2  sinh  £  (jr  +7)  cosh  ^  (x—y). 


sinh 


HYPERBOLIC  FUNCTIONS  77 

EXERCISES 

70.  (i.)  Prove  sinho=o,  cosho=i. 
(2.)  Prove  sinh  £TT/  =  /,  cosh^7r/=o. 
(3.)  Prove  sinh  TT/  =  O,  cosh7i7  =  —  i. 

Prove  that 

(4.)  si  n  (*—  /,r)  =  —  sin  ix. 

(  5  .)  cos  (—  ix)  =  cos  /Jr. 

(6.)  sinh(—  x)  =  —  sinh  x. 

(7.)  cosh(  —  x}  =  cosher. 

Remark.—  The  hyperbolic  tangent,  cotangent,  secant,  and  cosecant 
are  defined  by 

sinhjr  cosher 

tanh-r  = 


cosher  smh^r 

sech-r  =  —  \  —  ,  eschar  =  • 


t  \^*3\-,LM.    **    ^—         .         . 

cosher  sinner 

Prove  that 

(8.)  tan  (tx)  —  i  tanh  x. 

(9.)  coth  ( — x}  =  —  coth  x. 

(10.)  sech  (— .r)  =  sech  x. 

(n.)  coshajr  —  sinh3jr=:i. 

(12.)  sech  a.r-f- tan  ha.r=  i. 

(13.)  coth'.r  —  csch'jr  =  i . 

(14.)  sinh(.i- — y)  =  sinh^r  cosh y  —  cosher  sinh^. 

(15.)  cosh(.i- — _y)  =  cosher  coshj  — sinhjr 


(,6.)  coshi.r=v/l 

(17.)  sinhw  — sinhr/  =  2  cosh  \(u  +  v)  sinh  \  (u — v). 

(i  8.)  cosh  u  -\-  cosh  v  =  2  cosh  %(u-\-v)  cosh  \(u^v). 

(19.)  cosh  u  —  cosh  v  =  2  sin h^(#-f-?/)  sinh %(u  —  z/). 


CHAPTER  VII 

MISCELLANEOUS     EXERCISES 
RELATION   OF  FUNCTIONS 

77.  Prove  the  following  : 

(i.)  cos^r  =  sin^r  cot-r. 

(2.)  CSC.T  tan  x  =  sec  x. 

(3.)  (tan  x  -|-  cot  x)  s\nx  cos^r=:i. 

(4.)  (sec/  —  tan  /)  (sec  y  -f  tan  /)  =  i  . 

(5.)  (CSC  2  —  COt  2)  (CSC  2  -f-  COt  z)  =  I  . 

(6.)  cos2/  +  (tan  /  —  cot/)  sin/  cos  y  =  sin2/. 
(7.)  cos4.r  —  sin4*  -{-1=2  cos2,r. 
.)  (sinj/  —  cos/)2  =  1  —  2  sin  j  cos/. 

—  sin^r  COSJT). 


.  cot  x-\-  tan  y 

(10.)  -    —  -•  -  —  =  cot.  r  tan  y. 
tan  ^-  +  cot/ 

\-<n.)  cos2/  —  sin2/  =  2  cos2/—  i. 
(12.)  i  —  tan4^r  =  2  sec2^r  —  sec*;r. 


(i  3-)  -  -  5—  =  tan  jr. 
sm^r  cot2^: 

(14.)  sec2/  esc2/  =  tan2/  -f-  cot2/  +2. 

.)  cot/  —  esc/  sec/  (i  —  2  sin2/)  =  tan/. 


,    /     I  \2        I  —  COS.? 

(16.)  H  —  —  cot  2^)  = 

Vsin«  /       i  4-  cos  z 


I  +cos/  sin3/ 

(i  8.)  i+ 


(19.)  —  -  --  sin3.r  =  (cos-r  —  sin  x)  (i-|-sin.r  cos,r). 
(20.)  (sin,r  cos/-f-cos.r  sin/)2-j-(cos.r  cos/  —  sin  .r  si 


MISCELLANEOUS  EXERCISES  79 

(21.)  (a  cos.r  —  b  sin  .r)2 -(-(,*  sin  x  +  b  cos.r)2  =  &*+&*. 

-> '  ^    4tan2j 

— sin2jf  ~        (i—  tan2;/)2' 


Find  an  angle  not  greater  than  90°  which  satisfies  each  of  the  fol- 
lowing equations: 
(23.)  4  cos  x  =  3  sec  .r. 
(24.)  sin^^cscj  — f. 
(25.)   \/2  sin.r  —  tan.r  =  o. 
(26.)  2  cos.r—  \/3  cot.r  =  o. 
(27.)  tan  j  +  cot  j  —  2  =  0. 
(28.)  2  sin'-/  —  2  =  —  \/2  cosj. 
(29.)  3  tan2.r  —  i  =4  sin2.r. 
(30.)  cos2.r-|-2  sina.r  — -|  sin.r  =  o. 
(31.)  csc.r  =  §tan.r. 
(32.)  sec  .r  -(-  tan  .r  —  ±  \/3- 
(33.)  tan  .r  +  2  v/3  cos.r  =  o. 
(34.)  3  sin.r  —  2  cos2.r=o. 

Express  the  following  in  terms  of  the  functions   of  angles  less 
than  45°: 
(35.)  sin  92°. 
(36.)  cos  1 27°. 
(37.)  tan  320°. 
(38.)  cot  350°. 
(39.)  sin  2f>5  . 
(40.)  tan  171°. 

(41.)  Given  sin  .r  =  $  and  x  in  quadrant  II;  find  all  the  other 
functions  of  x. 

(42.)  Given  cos.n=  —  f  and  x  in  quadrant  III;  find  all  the  other 
functions  of  x. 

(43.)  Given  tan.r  =  |  and  x  in  quadrant  III;  find  all  the  other 
functions  of  x. 

.'44.)  Given  cot,r  =  — £  and  x  in  quadrant  IV;  find  all  the  other 
functions  of  x. 


8o  PLANE    TRIGONOMETRY 

In  what  quadrants  must  the  angles  lie  which  satisfy  each  of  the 
following  equations : 
(45.)  sin^r  cos.*'  —  i\/3. 
(46.)  sec.r  tan  .*•  =  2-^/3. 
(47.)  tanjj/  +  -y/20  cos}>  =  o.  - 
(48.)  cos  x  cot  x  =  £. 

Find  all  the  values  of  y  less  than  360°  which  will  satisfy  the  fol- 
lowing equations : 

(49.)  tanj-f-2  sinj  =  o. 

(50.)  (i  -f-  tan  x)  (i  —  2  sin  x]  =  o. 

(51.)  sin^r  COS.T  (i  +  2  cos-r)=o. 


Prove  the  following: 
(52.)  cos  780°  =  £. 

(53.)  sin  1485°  =  i\/2. 
(54.)  cos  2550°  =  ^  -y/3- 
(55.)  sin  (—  3000°)  =  —  cos  30°. 
(56.)  cos  1 300°  =  —  cos  40°** 

(57.)  Find  the  value  of  a  sin  90°  +  b  tano°-\-a  cos  180°. 
(58.)  Find  the  value  of  a  sin  30°  +  ^  tan  45°  -\-a  cos  60° -\-b  tan  135°. 
(59.)  Find  the  value  of  (a  —  b)  tan  225° +  £  cos  180°  —  a  sin  270°. 
(60.)  Find  the  value  of  (a  sin  45°+^  cos  45°)  (a  sin  135°  +  ^  sin  225°). 

RIGHT   TRIANGLES 

7£.  In  the  following  problems  the  planes  on  which  distances  are  measured 
are  understood  to  be  horizontal  unless  otherwise  stated. 

(i.)  The  angle  of  elevation  of  the  top  of  the  tower  from  a  point 
1 121  ft.  from  its  base  is  observed  to  be  15°  17';  find  the  height  of 
the  tower. 

(2.)  A  tree,  77  ft.  high,  stands  on  the  bank  of  a  river ;  at  a  point  on 
the  other  bank  just  opposite  the  tree  the  angle  of  elevation  of  the 
top  of  the  tree  is  found  to  be  5°  if  37".  Find  the  breadth  of  the 


MISCELLANEOUS  EXERCISES  81 

(3.)  What  angle  will  a  ladder  42  ft.  long  make  with  the  ground  if  its 
foot  is  25  ft.  from  the  base  of  the  building  against  which  it  is  placed  ? 

(4.)  When  the  altitude  of  the  sun  is  33°  22',  what  is  the  height  of  a 
tree  which  casts  a  shadow  75  ft.  ? 

(5.)  Two  towns  are  3  miles  apart.  The  angle  of  depression  of  one, 
from  a  balloon  directly  above  the  other,  is  observed  to  be  8°  15'. 
How  high  is  the  balloon  ? 

(6.)  From  a  point  197  ft.  from  the  base  of  a  tower  the  angle  of  ele- 
vation was  found  to  be  46°  45'  54" ;  find  the  height  of  the  tower. 

(7.)  A  man  5  ft.  10  in.  high  stands  at  a  distance  of  4  ft.  7  in.  from 
a  lamp-post,  and  casts  a  shadow  18  ft.  long;  find  the  height  of  the 
lamp-post. 

(8.)  The  shadow  of  a  building  101.3  ft-  high  is  found  to  be  131.5 
ft.  long;  find  the  elevation  of  the  sun  at  that  time. 

(9.)  A  rope  112  ft.  long  is  attached  to  the  top  of  a  building  and 
reaches  the  ground,  making  an  angle  of  77°  20'  with  the  ground ; 
find  the  height  of  the  building. 

(10.)  A  house  is  130  ft.  above  the  water,  on  the  banks  of  a  river; 
from  a  point  just  opposite  on  the  other  «eank  the  angle  of  elevation 
of  the  house  is  14°  30'  21".  Find  the  width  of  the  river. 

(u.)  From  the  top  of  a  headland,  1217.8  ft.  above  the  level  of  the 
sea,  the  angle  of  depression  of  a  dock  was  observed  to  be  10°  9'  13" ; 
find  the  distance  from  the  foot  of  the  headland  to  the  dock. 

(12.)  1121.5  ft.  from  the  base  of  a  tower  its  angle  of  elevation  is 
found  to  be  n°  3'  5  ";  find  the  height  of  the  tower. 

(13.)  One  bank  of  a  river  is  94.73  ft.  vertically  above  the  water,  and 
subtends  an  angle  of  10°  54'  13"  from  a  point  directly  opposite  at  the 
water's  edge;  find  the  width  of  the  river. 

(14.)  The  shadow  of  a  vertical  cliff  113  ft.  high  just  reaches  a  boat 
on  the  sea  93  ft.  from  its  base  ;  find  the  altitude  of  the  sun. 

(15.)  A  rope,  38  ft.  long,  just  reached  the  ground  when  fastened  to 
the  top  of  a  tree  29  ft.  high.  What  angle  does  it  make  with  the 
ground  ? 

(16:)  A  tree  is  broken  by  the  wind.  Its  top  strikes  the  ground  15 
ft.  from  the  foot  of  the  tree,  and  makes  an  angle  of  42°  28'  with  the 

ground.     Find  the  height  of  the  tree  before  it  was  broken. 
6 


82  PLANE    TRIGONOMETRY 

(17.)  The  pole  of  a  circular  tent  is  18  ft.  high,  and  the  ropes  reach- 
ing from  its  top  to  stakes  in  the  ground  are  37  ft.  long;  find  the 
distance  from  the  foot  of  the  pole  to  one  of  the  stakes,  and  the  angle 
between  the  ground  and  the  ropes. 

(i 8.)  A  ship  is  sailing  southwest  at  the  rate  of  8  miles  an  hour. 
At  what  rate  is  it  moving  south  ? 

(19.)  A  building  is  121  ft.  high.  From  a  point  directly  across  the 
street  its  angle  of  elevation  is  65°  3'.  Find  the  width  of  the  street. 

(20.)  From  the  top  of  a  building  52  ft.  high  the  angle  of  elevation 
of  another  building  112  ft.  high  is  30°  12'.  How  far  are  the  buildings 
apart  ? 

(21.)  A  window  in  a  house  is  24  ft.  from  the  ground.  What  is  the 
inclination  of  a  ladder  placed  8  ft.  from  the  side  of  the  building  and 
reaching  the  window  ? 

(22.)  Given  that  the  sun's  distance  from  the  earth  is  92,000,000 
miles,  and  its  apparent  semidiameter  is  16'  2"  ;  find  its  diameter. 

(23.)  Given  that  the  radius  of  the  earth  is  3963  miles,  and  that  it 
subtends  an  angle  of  57'  2"  at  the  moon;  find  the  distance  of  the 
•moon  from  the  earth. 

(24.)  Given  that  when  the  moon's  distance  from  the  earth  is  238885 
miles,  its  apparent  semidiameter  is  15' 34";  find  its  diameter  in  miles. 

(25.)  Given  that  the  radius  of  the  earth  is  3963  miles,  and  that  it 
subtends  an  angle  of  9"  at  the  sun ;  find  the  distance  of  the  sun 
from  the  earth. 

(26.)  A  light-house  is  57  ft.  high  ;  the  angles  of  elevation  of  the  top 
and  bottom  of  it,  as  seen  from  a  ship,  are  5°  3'  20"  and  4°  28'  8".  Find 
the  distance  of  its  base  above  the  sea-level. 

(27.)  At  a  certain  point  the  angle  of  elevation  of  a  tower  was  ob- 
served to  be  53°  51'  1 6",  and  at  a  point  302  ft.  farther  away  in  the 
same  straight  line  it  was  9°  52'  10";  find  the  height  of  the  tower. 

(28.)  A  tree  stands  at  a  distance  from  a  straight  road  and  between 
two  mile-stones.  At  one  mile-stone  the  line  to  the  tree  is  observed 
to  make  an  angle  of  25°  15'  with  the  road,  and  at  the  other  an  angle 
of  45°  17'.  Find  the  distance  of  the  tree  from  the  road. 

(29.)  From  the  top  of  a  light-house,  225  ft.  above  the  level  of  the 
sea,  the  angle  of  depression  of  two  ships  are  17°  21'  50"  and  13°  50'  22", 


MISCELLANEOUS  EXERCISES  83 

and  the  line  joining  the  ships  passes  directly  beneath  the  light-house ; 
find  the  distance  between  the  two  ships. 

ISOSCELES   TRIANGLES  AND    REGULAR   POLYGONS 

S^- 
79.  (i.)  The  area  of  a  regular  dodecagon  is  37.52  ft.;   find    its 

apothem. 

(2.)  The  perimeter  of  a  regular  polygon  of  1 1  sides  is  23.47  ft. ;  find- 
the  radius  of  the  circumscribing  circle. 

(3.)  A  regular  decagon  is  circumscribed  about  a  circle  whose  radius 
is  3.147  ft. ;  find  its  perimeter. 

(4.)  The  side  of  a  regular  decagon  is  23.41  ft. ;  find  the  radius  of 
the  inscribed  circle. 

(5.)  The  perimeter  of  an  equilateral  triangle  is  17.2  ft.;  find  the 
area  of  the  inscribed  circle. 

(6.)  The  area  of  a  regular  octagon  is  2478  sq.  in. ;  find  its  pe- 
rimeter. 

(7.)  The  area  of  a  regular  pentagon  is  32.57  sq.  ft. ;  find  the  radius 
of  the  inscribed  circle. 

(8.)  The  angle  between  the  legs  of  a  pair  of  dividers  is  43°,  and  the 
legs  are  7  in.  long ;  find  the  distance  between  the  points. 

(9.)  A  building  is  37.54  ft.  wide,  and  the  slope  of  the  roof  is  43°  36' ; 
find  the  length  of  the  rafters. 

(10.)  The  radius  of  a  circle  is  12732,  and  the  length  of  a  chord  is 
18321 ;  find  the  angle  the  chord  subtends  at  the  centre. 

(u.)  If  the  radius  of  a  circle  is  taken  as  unity,  what  is  the  length 
of  a  chord  which  subtends  an  angle  of  77°  17'  40"? 

(12.)  What  angle  at  the  centre  of  a  circle  does  a  chord  which  is  ^ 
of  the  radius  subtend  ? 

(13.)  What  is  the  radius  of  a  circle  if  a  chord  11223  ft.  subtends  an 
angle  of  59°  50'  52"? 

(14.)  Two  light-houses  at  the  mouth  of  a  harbor  are  each  2  miles 
from  the  wharf.  A  person  on  the  wharf  finds  the  angle  between  the 
lines  to  the  light-houses  to  be  17°  32'.  Find  the  distance  between  the 
two  light-houses. 

(15.)  The  side  of  a  regular  pentagon  is  2;  find  the  radius  of  the 
inscribed  circle. 


84  PLANE    TRIGONOMETRY 

(i 6.)  The  perimeter  of  a  regular  heptagon  inscribed  in  a  circle  is 
12  ;  find  the  radius  of  the  circle. 

(17.)  The  radius  of  a  circle  inscribed  in  an  octagon  is  3;  find  the 
perimeter  of  the  octagon. 

(18.)  A  regular  polygon  of  9  sides  is  inscribed  in  a  circle  of  unit 
radius;  find  the  radius  of  the  inscribed  circle. 

(19.)  Find  the  perimeter  of  a  regular  decagon  circumscribed  about 
a  unit  circle. 

(20.)  Find  the  area  of  a  regular  hexagon  circumscribed  about  a 
unit  circle. 

r      (21.)  Find  the  perimeter  of  a  polygon  of  11  sides  inscribed  in  a 
\  unit  circle. 

(22.)  The  perimeter  of  a  dodecagon  is  30;  find  its  area. 

(23.)  The  area  of  a  regular  polygon  of  11  sides  is  18;  find  its  pe- 
rimeter. 

TRIGONOMETRIC    IDENTITIES   AND   EQUATIONS 

8O.  Prove  the  following : 
(i.)  sin 


sin  2x  -f-  si 


(4.)  cos2/  tan2/  -f  sin2/  cot8/  =  I. 

(5.)  -.  —  =  cot  x  cot  y  cot  z  —  cot  x  —  cot  y  —  cot  z. 

sin*  sin/  sin  z 

(6.)  cos2  (x  —y)  —  sin2  (x  +/)  =  cos  2  x  cos  2/. 

.    sin  jr-4-siny 

(7.)  -  —  —  =  —  cot  i  (x  —  y). 

cos  x  —  cos  y 

.  cosx  —  sec^r 


sin  2x 

(o.)  cot;r  =  — 

i —  cos  2x 

I  —  COS  2/ 

(10.)  tan  y  =  —        — —  • 
i  +  cos  2/ 

(11.)  cot  x  —  tan  .r  =  2  cot  2x.  v, 


/ 


MISCELLANEOUS  EXERCISES  85 


(12.)  tan^.r  +  2  sin2  ^,r  cot.r 

tan  x  ±  tan  / 

(n.)  -  —  —  =  dbsm^r  sec.r  tan/. 

cot  .r±  cot/ 

(14.)  sin  .r  —  2  sin3  ,r  =  sin  x  cos  2.r. 

(15.)  4  sin/  sin  (60°  —  /)  sin  (60°  -(-/)  =  sin  3/. 

suy(,-tan^)/      _1_   _     -  '  =  si 

sec2/  Vcos/  —  sin/     cos/+sin/ 


(17.)  i  +  tan/  tan  £/= 

(1  8.)  sin  4^r  =  4  sin.r  cos3  ^r  —  4  cos^r  si 


(20.)  tan  50°  4-  cot  50°  —  2  sec  10°. 
(21.)  cos  (x  +  45°)  4-  sin  (x  —  45°)  =  o. 
tan^r 


(22.) 


i  —  cot  2x  tan  x 


(23.)  (  i  —  tan2  x)  sin  x  cos  .r  =  cos  2.r  yj 


—  cos  2x 

-f  COS  2X  ' 


-=  ec 

'  cos/  —  sin/ 


(25.)  sin  (*+/)  cos,r—  cos(.r+/)  sin^r  =  si 
(26.)  cos  (.r  —  /)  sin/  +  sin  (x  —  /)  cos/  =  sin  JJT. 
.    sin(jr—  /)        sin(/—  g)  ,    sin  (g  —  JT)  _ 

(27  )    -  _  _i_  -  --  --4-  —  —  —  u. 

COSX   COS/        COS/   COS.?        COS  2  COSX 

sin£+sirL2£  =  co 
cos  .r  —  cos  2.r 

(29.)  2  sin2  ,r  sina/-h  2  cos2  x  cos2/  =  i  +  cos  2x  cos  2/. 
(30.)  sin  60°  4-  sin  30°  =  2  sin  45°  cos  15°. 

tan  (,r-/)  +  tan/ 


u  ';  i—  tan  (.r—,/)  tan/ 


sin/  tan  i/ 

(33.)  sin^r+sin  2.r  =  2  sin 
sin  x  4-  sin/  _ 


cos  .*•  —  cos/      sn/  —  sn  x 


(36.)  2  tan  2/  =  tan(45°+/)  —  tan  (45°—  /). 


86  PLANE    TRIGONOMETRY 

tan2.r4-tan.r  _sin3.r 
tan  2  x  —  tan  x  ~  sin^r 

(38.) 


i  —  3  tan2/ 

(39.)  sin  60°  4-  sin  20°  =  2  sin  40°  cos  20°. 
(40.)  sin  40°  —  sin  10°  =  2  cos  2  5°  sin  15°. 
(41.)  cos  ix  —  cos4-r  =  2  sin  yc  sin  x. 
(42.)  tan  15°  =  2—  v/3- 
(43.)  (\/i  4sin.r  —  \/i  —  sin.r)2=4  sin2 
(44.)     "V/i  H-sin^r--'i  —  sin^)2  =  4  cos 


,  ...  N  siii4.r 

(46.)       .     *      =  2  COS  2.T. 


(47.)  sin  50°  —  sin7o°-f-sinio°  =  o. 

f.Q\  I?  IT  ^TT     .       IT 

(40-)  cos  --  cos-  =  2  sin  —sin  —  • 

3  J2  12          12 


sin  750  -  sin  1  5°          /f 
y  cos75°4-cosi5°     V     ' 


(51.)  tan8£.r(i+cota£.r)3=  --— 


(52.)  tan  7  5°  =  2 

(53.)  sin  3^-  -f-  sin  5.1-  =  2  sin  4^-  cos^r. 

(54.)  cos  5-r  -f-  cos  9-r  =  2  cos  7-r  cos  2^r. 

~  1 


(55.)  sin  1  5°  = 


\/2 


(56.)  -  -  =tan;r. 

cos  3-r  +  cos  ^* 

(57.)  sin  5j  =  5  sinj  —  20  sin3/-}-  16  sin5/. 
(58.)  cos5/  =  5  cosj  —  20  cos3/  +  16  cos5/. 


(60.) 

(6  1  .)  cos  3^-  -f-  cos  5a-  -}-  cos  7*  +  cos  1  5^-  =  4  cos  4^  cos  5  ^r  cos  £>x 


(62.)  sin2  \x  (cot  \.  i-  —  i)2=  i  —  sin.r. 
—  sin  \r 


MISCELLANEOUS  EXERCISES  87 

sin2  \x  (c 
(63.) 

(64.) 

4 

(65.) 

cos  .r  —  sin.r  2 

(66.)  cosj  +  cos  I1  2O  —y}  +  cos  (  1  2 

x  si  113.1- 

(67.)      .        =2  cos2.r+  1. 
sin.r 

(68) 


. 

cos  3-r  4-  3  cos  x 

(64.)  sin.r(i-|-tan.r)4-cos.r(i  4-cot.r)  =  esc  ^  +  sec  ^r. 
cos3.r  —  sin3.r 


(sin  y  — sinj)(cos4_y  — cos6y) 
i  —  cos.r 


4cos.r 


,  , 

(70.)   —-  —  -  2-  =  2. 

sin.r        cos.r 

s    i  H-  sin  a-  4-  cos.  r 
(71.)  -  —  =cotfr. 

i  +sm.r  —  cos.r 


cos  (4-r  —  27)  4-  cos  (4.T  — 
Sin.r  +  sin3.r  +  sin5.r4-sin7.r  =  ^ 
cos  .r  -f-  cos  3.r  -|-  cos  $.r  4-  cos  7.1- 

If  A,  B,  and  C  are  the  angles  of  a  triangle,  prove  the  following 
(74.)  sin2^4-|-sin2/>'4-sin2C  =  4  sin  A  s\n£  sinC 
(75.)  sin  2^f  +  sin  27)'  —  sin  2(7  =  4  cos^4  cos#  sinC. 
(76.)  sinM4-sin2/»'-f-sin2C=2  +  2  cos  A  cos  B  cos  C 
(77.)  tan  ^  -f  tan  B  4-  tan  C  =  tan  A  tan  ^  tan  C. 

Solve  the  following  equations  for  values  of  x  less  than  360°. 

(78.)  cos  2.r  4-  cos  x  =  —  i  . 

(79.)  sin.r4-sin7.r 

(80.)  cos^"  —  sin2.r  — 

(8  1.)  cos.r  —  sin3^r  —  cos2.i-  =  o. 

(82.)  sin^a-  —  2  sin2,r  r=o. 

(83.)  sin  2.r  —  cos  2.r  —  sin  x  +  cos  x  =  o. 

(84.)  sin  (60°  —  .r)  -  sin  (60°  +  x)  =  +  i  y 

(85.)  sin  (30°  4-  -r)  —  cos  (60°  +  x)  =  —  |  ^ 


88  PLANE    TRIGONOMETRY 

(86.)  esc  x  =  i  +  cot  x. 

(87.)  cos  T.X  =  cos  2.r. 

(88.)  2  sin_y=sin  2y. 

(89.)  sin  $y -\- sin  2y -\- s'my  =  o. 

(90.)  sina.r  +  5  cos'.r  =  3. 

(91.)  tan(45°  — 


OBLIQUE   TRIANGLES 

81.  (i.)  It  is  required  to  find  the  distance  between  two  points,  A 
and  JB.on  opposite  sides  of  a  river.  A  line,  AC,  and  the  angles  BAG 
and  ACB  are  measured  and  found  to  be  2483  ft.,  61°  25',  and  52°  17' 
respectively. 

°(2.)  A  straight  road  leads  from  a  town  A  to  a  town  B,  12  miles 
distant ;  another  road,  making  an  angle  of  77°  with  the  first,  goes  from 
A  to  a  town  C,  7  miles  distant.  How  far  are  the  towns  B  and  C  apart  ? 
In  order  to  determine  the  distance  of  a  fort,  A,  from  a  battery, 
B,  a  line,  BC,  one-half  mile  long,  is  measured,  and  the  angles  ABC 
and  ACB  are  observed  to  be  75°  18'  and  78°  21'  respectively.  Find 
the  distance  AB. 

(4.)  Two  houses,  A  and  B,  are  1728  ft.  apart.  Find  the  distance  of 
a  third  house,  C,  from  A  if  BAC=tf°  51 'and  ABC=  57°  23'. 

(5.)  In  order  to  determine  the  distance  of  a  bluff,  A,  from 'a  house, 
B,  in  a  plane,  a  line,  BC,  was  measured  and  found  to  be  1281  yards, 
also  the  angles  ABC  and  BCA  65°  31'  and  70°  2'  respectively.  Find 
the  distance  AB. 

(6.)  Two  towns,  3  miles  apart,  are  on  opposite  sides  of  a  balloon. 
The  angles  of  elevation  of  the  balloon  are  found  to  be  13°  19'  and 
20°  3'.  Find  the  distance  of  the  balloon  from  the  nearer  town. 

(7.)  It  is  required  to  find  the  distance  between  two  posts,  A  and  B, 
which  are  separated  by  a  swamp.  A  point  C  is  1272.5  ft.  from  A,  and 
2012.4  ft-  from  B.  The  angle  ACB  is  41°  9'  1 i". 

(8.)  Two  stakes,  A  and  B,  are  on  opposite  sides  of  a  stream  ;  a 
third  point,  C,  is  so  situated  that  the  distances  AC  and  BC  can  be 
found,  and  are  431.27  yards  and  601.72  yards  respectively.  The  angle 
ACB  is  39°  53'  13".  Find  the  distance  between  the  stakes  A  and  B. 


MISCELLANEOUS   EXERCISES  89 

(9.)  Two  light-houses,  A  and  B,  are  11  miles  apart.  A  ship,  C,  is 
observed  from  them  to  make  the  angles  BAC  =31°  13'  31"  and  ABC 
=  21°  46'  8".  Find  the  distance  of  the  ship  from  A. 

(10.)  Two  islands,  A  and  B,  are  6103  ft.  apart.  Find  the  distance 
from  A  to  a  ship,  C,  if  the  angle  ABC  is  37°  25'  and  BAC  is  40°  32'. 

(u.)  In  ascending  a  cliff  towards  a  light-house  at  its  summit,  the 
light-house  subtends  at  one  point  an  angle  of  21°  22'.  At  a  point 
55  ft.  farther  up  it  subtends  an  angle  of  40°  27'.  If  the  light-house 
is  58  ft.  high,  how  far  is  this  last  point  from  its  foot? 

(12.)  The  distances  of  two  islands  from  a  buoy  are  3  and  4  miles 
respectively.  The  islands  are  2  miles  apart.  Find  the  angle  sub- 
tended by  the  islands  at  the  buoy. 

^•(13.)  The  sides  of  a  triangle  are  151.45,  191.32,  and  250.91.  Find 
the  length  of  the  perpendicular  from  the  largest  angle  upon  the 
opposite  side. 

**  (14.)  A  tree  stands  on  a  hill,  and  the  angle  between  the  slope  of  the 
hill  and  the  tree  is  110°  23'.  At  a  point  85.6  ft.  down  the  hill  the 
tree  subtends  an  angle  of  22°  22'.  Find  the  height  of  the  tree. 
!  (15.;  A  light-house  54  ft.  high  is  built  upon  a  rock.  From  the  top 
of  the  light-house  the  angle  of  depression  of  a  boat  is  19°  10',  and 
from  its  base  the  angle  of  depression  of  the  boat  is  12°  22'.  Find  the 
height  of  the  rock  on  which  the  light-house  stands. 

(16.)  Three  towns,  A,  B,  and  C,  are  connected  by  straight  roads. 
A£  =  4  miles,  BC=  5  miles,  and  AC=  7  miles.  Find  the  angle  made 
by  the  roads  AB  and  BC. 

(17.)  Two  buoys,  A  and  B,  are  one-half  mile  apart.  Find  the  dis- 
tance from  A  to  a  point  C  on  the  shore  if  the  angles  ABC  and  BAC 
are  77°  7'  and  67°  17'  respectively. 

(i 8.)  The  top  of  a  tower  is  175  ft.  above  the  level  of  a  bay.  From 
its  top  the  angles  of  depression  of  the  shores  of  the  bay  in  a  certain 
direction  are  57°  16'  and  15°  2'.  Find  the  distance  across  the  bay. 

(19.)  The  lengths  of  two  sides  of  a  triangle  are  \/2  and  -v/3-  The 
angle  between  them  is  45°.  Find  the  remaining  side. 

(20.)  The  sides  of  a  parallelogram  are  172.43  and  101.31,  and  the 
angle  included  by  them  is  61°  16'.  Find  the  two  diagonals. 

(21.)  A  tree  41  ft.  high  stands  at  the  top  of  a  hill  which  slopes 


X 

X 


90  PLANE   TRIGONOMETRY 

10°  12'  to  the  horizontal.  At  a  certain  point  down  the  hill  the  tree 
subtends  an  angle  of  28°  29'.  Find  the  distance  from  this  point  to 
the  toot  of  the  tree. 

(22.)  A  plane  is  inclined  to  the  horizontal  at  an  angle  of  7°  33'.  At 
a  certain  point  on  the  plane  a  flag-pole  subtends  an  angle  20°  3',  and  at 
a  point  50  ft.  nearer  the  pole  an  angle  of  40°  35'.  Find  the  height  of 
the  pole. 

(23.)  The  angle  of  elevation  of  an  inaccessible  tower,  situated  in  a 
plane,  is  53°  19'.  At  a  point  227  ft.  farther  from  the  tower  the  angle 
of  elevation  is  22°  41'.  Find  the  height  of  the  tower. 

(24.)  A  house  stands  on  a  hill  which  slopes  12°  1 8'  to  the  horizontal. 
75  ft.  from  the  house  down  the  hill  the  house  subtends  an  angle  of 
32°  5'.  Find  the  height  of  the  house. 

(25.)  From  one  bank  of  a  river  the  angle  of  elevation  of  a  tree  on 
the  opposite  bank  is  28°  31'.  From  a  point  139.4  ft.  farther  away  in  a 
direct  line  its  angle  of  elevation  is  19°  10'.  Find  the  width  of  the  river. 

(26.)  From  the  foot  of  a  hill  in  a  plane"  the  angle  of  elevation  of 
the  top  of  the  hill  is  21°  7'.  After  going  directly  away  211  ft.  farther, 
the  angle  of  elevation  is  18°  37'.  Find  the  height  of  the  hill. 

(27.)  A  monument  at  the  top  of  a  hill  is  153.2  ft.  high.  At  a  point 
321.4  ft.  down  the  hill  the  monument  subtends  an  angle  of  11°  13'. 
Find  the  distance  from  this  point  to  the  top  of  the  monument. 

(28.)  A  building  is  situated  on  the  top  of  a  hill  which  is  inclined 
10°  12'  to  the  horizontal.  At  a  certain  distance  up  the  hill  the  angle 
of  elevation  of  the  top  of  the  building  is  20°  55',  and  115.3  ft-  farther 
down  the  hill  the  angle  of  elevation  is  15°  10'.  Find  the  height  of 
.the  building. 

(29.)  A  cloud,  C,  is  observed  from  two  points,  A  and  J3,  2874  ft. 
apart,  the  line  AB  being  directly  beneath  the  cloud.  At  A,  the  angle 
of  elevation  of  the  cloud  is  77°  19',  and  the  angle  CAB  is  51°  18'. 
The  angle  ABC  is  found  to  be  60°  45'.  Find  the  height  of  the  cloud 
above  A. 

(30.)  Two  observers,  A  and  B,  are  on  a  straight  road,  675.4  ft.  apart, 
directly  beneath  a  balloon,  C.  The  angles  ABC  and  BAC  are  34°  42' 
and  41°  15'  respectively.  Find  the  distance  of  the  balloon  from  the 
first  observer. 


MISCELLANEOUS  EXERCISES  91 

(31.)  A  man  on  the  opposite  side  of  a  river  from  two  objects,  A 
and  B,  wishes  to  obtain  their  distance  apart.  He  measures  the  dis- 
tance CD  =  357  ft.,  and  the  angles  ACB=2^>  33',  BCD  =  38°  52',  ADB 
=  54°  10',  and  ADC =34°  n'.  Find  the  distance  AB. 
i  (32.)  A  cliff  is  327  ft.  above  the  sea-level.  From  the  top  of  the 
cliff  the  angles  of  depression  of  two  ships  are  15°  11'  and  13°  13'. 
From  the  bottom  of  the  cliff  the  angle  subtended  by  the  ships  are 
122°  39'.  How  far  are  the  ships  apart  ? 

(33.)  A  man  standing  on  an  inclined  plane  112  ft.  from  the  bottom 
observed  the  angle  subtended  by  a  building  at  the  bottom  to  be  33° 
52'.  The  inclination  of  the  plane  to  the  horizontal  is  18°  51'.  Find 
the  height  of  the  building. 

(34)  Two  boats,  A  and  B,  are  451.35  ft.  apart.  The  angle  of  ele- 
vation of  the  top  of  a  light-house,  as  observed  from  A,  is  33°  if. 
The  base  of  the  light-house,  C,  is  level  with  the  water;  the  angles 
ABC  and  CAB  are  12°  31'  and  137°  22'  respectively.  Find  the  height 
of  the  light-house. 

(35.)  From  a  window  directly  opposite  the  bottom  of  a  steeple  the 
angle  of  elevation  of  the  top  of  the  steeple  is  29°  21'.  From  another 
window,  20  ft.  vertically  below  the  first,  the  angle  of  elevation  is  39°  3'. 
Find  the  height  of  the  steeple. 

(36.)  A  dock  is  i  mile  from  one  end  of  a  breakwater,  and  i£  miles 
from  the  other  end.  At  the  dock  the  breakwater  subtends  an  angle 
of  31°  n'.  Find  the  length  of  the  breakwater  in  feet. 

(37.)  A  straight  road  ascending  a  hill  is  1022  ft.  long.  The  hill 
rises  i  ft.  in  every  4.  A  tower  at  the  top  of  the  hill  subtends  an 
angle  of  7°  19'  at  the  bottom.  Find  the  height  of  the  tower. 

(38.)  A  tower,  192  ft.  high,  rises  vertically  from  one  corner  of  a 
triangular  yard.  From  its  top  the  angles  of  depression  of  the  other 
corners  are  58°  4'  and  17°  49'.  The  side  opposite  the  tower  subtends 
from  the  top  of  the  tower  an  angle  of  75°  15'.  Find  the  length  of 
this  side. 

(39.)  There  are  two  columns  left  standing  upright  in  a  certain  ruins  ; 
the  one  is  66  ft.  above  the  plain,  and  the  other  48.  In  a  straight  line., 
between  them  stands  an  ancient  statue,  the  head  of  which  is  100'  ft. 
from  the  summit  of  the  higher,  and  84  ft.  from  the  top  of  the  lower 


92  PLANE    TRIGONOMETRY 

column,  the  base  of  which  measures  just  74  ft.  to  the  centre  of  the 
figure's  base.  Required  the  distance  between  the  tops  of  the  two 
columns. 

(40.)  Two  sides  of  a  triangle  are  in  the  ratio  of  1 1  to  9,  and  the 
opposite  angles  have  the  ratio  of  3  to  i.  What  are  these  angles  ? 

(41.)  The  diagonals  of  a  parallelogram  are  12432  and  8413,  and  the 
angle  between  them  is  78°  44' ;  find  its  area. 

(42.)  One  side  of  a  triangle  is  1012.6  and  two  angles  are  52°  21'  and 
57°  32' ;  find  its  area. 

(43.)  Two  sides  of  a  triangle  are  218.12  and  123.72,  and  the  included 
angle  is  59°  10' ;  find  its  area. 

(44.)  Two  angles  of  a  triangle  are  35°  15'  and  47°  18',  and  one  side 
is  2104.7  I  find  its  area. 

(45.)  The  three  sides  of  a  triangle  are  1.2371,  1.4713,  and  2.0721; 
find  the  area. 

(46.)  Two  sides  of  a  triangle  are  168.12  and  179.21,  and  the  included 
angle  is  41°  14' ;  find  its  area. 

(47.)  The  three  sides  of  a  triangle  are  51  ft.,  48.12  ft.,  and  32.2  ft. ; 
find  the  area. 

(48.)  Two  sides  of  a  triangle  are  1 1 1 . 1 8  and  121.21,  and  the  included 
angle  is  27°  50' ;  find  its  area. 

(49.)  The  diagonals  of  a  parallelogram  are  37  and  51,  and  they  form 
an  angle  of  65° ;  find  its  area. 

(50.)  If  the  diagonals  of  a  quadrilateral  are  34  and  56,  and  if  they 
intersect  at  an  angle  of  67°,  what  is  the  area  ? 


SPHERICAL  TRIGONOMETRY 


CHAPTER   VIII 

RIGHT  AND   QUADRANTAL  TRIANGLES 
RIGHT   TRIANGLES 

82.  Let  O  be  the  centre  of  a  sphere  of  unit  radius,  and 
ABC  a  right  spherical  triangle,  right  angled  at  A,  formed  by 
the  intersection  of  the  three  planes  A OC,  AOB,  and  BOC 


with  the  surface  of  the  sphere.  Suppose  the  planes  DAC" 
and  BEC  passed  through  the  points  A  and  B  respectively, 
and  perpendicular  to  the  line  OC.  The  plane  angles  DC" A 
and  BC'E  each  measure  the  angle  C  of  the  spherical  tri- 
angle, and  the  sides  of  the  spherical  triangle  a,  b,  c  have  the 
same  numerical  measure  as  BOC,  AOC,  and  AOB  respec- 


94  SPHERICAL    TRIGONOMETRY 


tively,  then,  AD  =  ta.nc,  BE  —  s\\\c,  BC1  =  sma, 

cos6,  OE  =  cosc,  AC"  =  sin  b. 
In  the  two  similar  triangles  OEC'  and  OAC"  ', 


OA         i  .      cos  b  '  ' 

•-  LOS  6*   LOS  6. 

V) 

In  the  triangle  BC'  E, 
^.n        BE    Qr  ^.n 

sin^r             v 

(2) 

^C?"0 
In  the  triangle  DAC"  , 

sin  a 

DA     or 

tan  ^ 

(3) 
(4) 

Combining  formulas  (2)  and  (3)  with 
.,     tan  $ 

(i). 

rrb^ 

tan  a 

Again,  if  AB  were  made  the  base  of  the  right  spherical 
triangle  ABC,  we  should  have 

sinj5=£^-  (5) 

4-i, •>    A 

(6) 

£r  (7) 

From   the  foregoing  equations  we  may  also  obtain   by 
combinations, 

cos/?=sin  C  cos^.  (8) 

cosC— sin  B  cose.  (9) 

cos  #  =  cot  B  cot  £7.  (10) 

NAPIER'S  RULES  OF  CIRCULAR  PARTS 
S3.  The  above  ten   formulas  are  sufficient  to  solve  all 
cases  of  right  spherical  triangles.     They  may,  however,  be 


RIGHT  AND   QUADRANTAL    TRIANGLES 


95 


expressed  as  two  simple   rules,  called,  after  their  inventor, 
Napier's  rules. 

The  two  sides  adjacent  to  the  right  angle,  the  complement 
of  the  hypotenuse,  and  the  complements  of  the  oblique  an- 
gles are  called  the  circular  parts. 

The  right  angle  is  not  one  of  the  circular  parts. 


comp  B 


comp 


comp  C 


Thus  there  are  fire  circular  parts — namely,  />,  c,  comprt,  comp/?,  compC 
Any  one  of  the  five  parts  may  be  called  the  middle  part,  then  the  two  parts  next 
to  it  are  called  adjacent  parts,  and  the  remaining  two  parts  are  called  the  oppo- 
site parts. 

Thus  if  c  is  taken  for  the  middle  part,  comp/?  and  b  are  adjacent  parts,  and 
comptf  and  comp  C  are  opposite  parts. 

The  ten  formulas  may  be  written  and  grouped  as  follows  : 


ist  Group. 

sin  comp  C  =  tan  comprt  tan  b. 
sin  comp  /?=  tan  compi?  tan  c. 
MII  comp  a  =  tan  comp/?  tan  comp  C. 
sin  c  =  tan  comp  B  tan  b. 

sin 


b  =tan  comp  C  tan  r. 


zd  Group. 

sin  comp  rt=r  cos  l>  cos  c. 
sin  b = cos  comp  a  cos  comp  B. 

sin  r=cos  comprt  cos  comp  C. 

sin  comp  j9=cos  comp  C  cos  £. 
sin  comp  f=cos  comp/?  cose-. 


Napier's  rules  may  be  stated  : 

I.  The  sine  of  the  middle  part  is  equal  to  the  product  of 
the  tangents  of-  the  adjacent  parts. 

II.  The  sine  of  the  middle  part  is  equal  to  the  product  of 
the  cosines  of  the  opposite  parts. 


96 


SPHERICAL  TRIGONOMETRY 


84.  In  the  right  spherical  triangles  considered  in  this  work,  each 
side  is  taken  less  than  a  semicircumference,  and  each  angle  less  than 
two  right  angles. 

In  the  solution  of  the  triangles,  it  is  to  be  observed, 

(i.)  If  the  two  sides  about  the  right  angle  are  both  less  or  both 
greater  than  90°,  the  hypotenuse  is  less  than  90°;  if  one  side  is  less 
and  the  other  greater  than  90°,  the  hypotenuse  is  greater  than  90°. 

(2.)  An  angle  and  the  side  opposite  are  either  both  less  or  both 
greater  than  90°. 


EXAMPLE 


85.  Given  #  =  63°  56',  £  =  40°  o',  to  find  c,  B,  and  C. 


To  find  c. 

cotnp  a  is  the  middle  part. 
c  and  b  are  the  opposite  parts, 
sin  comp  a=cos  b  cos  c, 
cos  0=cos  b  cos  c. 

cos  a 

cos  c  = -• 

cos  b 

log  cos  0=9.64288 
colog  cos  £=0.11575 
log  cos  ^-=9.75863 
'=54°  59  47" 


To  find  C. 

comp  C  is  the  middle  part. 

comp  a,  and  b  are  adjacent  parts. 

sin  comp  C=tan  comp  a  tan£, 

cos  C=  cot  a  tan£. 

log  cot  a= 9. 68946 
log  tan  £=9  92381 

9-61327 
C=65°  45'  58" 


To  find  B. 

l>  is  the  middle  part. 

comp  a  and  comp  B  are  the  opposite 

parts. 

sin  £=cos  comp  a  cos  comp  B, 
or  sin  £=sin  a  sin  B. 

•  sin  b 


log  sin  £=9.80807 
colog  sin  rt=o.  04659 
log  sin  £=9.  85466 
^  =  45°  41  '28" 

Check. 
Use  the  three  parts  originally  required. 

comp  C  is  the  middle  part. 
comp.#  and  c  are  opposite  parts. 

sin  comp  C=cosc  cos  comp  B, 
or  cos  C=cos  c  sin  B, 

log  cos  £-=9.75863 
log  sin  B=<).  85466 
log  cos  (7=9.61329 

C=6$°  45'  54" 


RIGHT  AND  QUADRANTAL    TRIANGLES  97 

AMBIGUOUS   CASE 

86.  When  a  side  about  the  right  angle  and  the  angle  opposite 
this  side  are  given,  there  are  two  solutions,  as  illustrated  by  the  fol- 
lowing figure.  Since  the  solution  gives  the  values  of  each  part  in 
terms  of  the  sine,  the  results  are  not  only  the  values  of  a,  b,  B,  but 
1 8o°— rt,  180°— b,  1 8o°— #. 


Given  c  =  26°  4'. 


To  find  a,  a',  b,  b'  and  B,  B',  using  Napier's  rules. 


To  find  B  and  B '. 

sin  comp  C=  cos  comp  B  cose, 
cos  C=sin  B  cos  c, 

cos  C 
~~  cos  c 

log  cos  C=g.  90796 

colog  cos  £-=0.04659 

log  sin ^  =  9.95455 

B=  64°  14'  30" 
r  =  i8o°-£=ii5°  45'  30" 

To  find  b  and  b' . 
sin  £=tan  c  tan  comp  C, 
sin  £=tan  c  cot  C 
log  tan  ^-=9.68946 
log  cot  67=0.13874 
log  sin  £=9.82820 

b—  42°  19'  17" 
£=137°  40'  43" 


To  find  a  and  a'. 
sin  £-=cos  comp  a  cos  Comp  C, 
sin  c=sin  a  sin  (7, 
sin  c 


or 
or 

log  sin  £-  =  9.64288 

colog  sin  (7=0.23078 

log  sin  0=9.87366 

a=  48°  22'  55"- 
fl'  =  i8o°-a=i3i°37'  5"  + 
'Discrepancy  due  to  omitted  decimals.) 

Check. 

sin  £=cos  comp  a  cos  comp  /?, 
or  sin  £=sin  a  sin  ^. 

log  sin  a  or  a'=g.  87366 

log  sin  #  or  .#'=9.95455 

log  sin  £=9.82821 

£=  42°  19'  21" 
39" 


98  SPHERICAL    TRIGONOMETRY 

QUADRANTAL   TRIANGLES 

87.  Def.  —  A  quadrantal  triangle  is  a  spherical  triangle 
one  side  of  which  is  a  quadrant. 

A  quadrantal  triangle  may  be  solved  by  Napier's  rules  for 
right  spherical  triangles  as  follows  : 

By  making  use  of  the  polar  triangle  where 


C=i8o°  —  ^  <r=i8o°—  C' 

we  see  that  the  polar  triangle  of  the  quadrantal  triangle  is 
a  right  triangle  which  can  be  solved  by  Napier's  rules. 
Whence  we  may  at  once  derive  the  required  parts  of  the 
quadrantal  triangle. 

EXAMPLE 

Given       A  =  1  36°  4'.             B  =  1  40°  o'.  a  —  90°  o'. 
The  corresponding  parts  of  the  polar  triangle  are 

a'  =^3°  56',            V  =  40°  o',  A'  =  90°. 
By  Napier's  rules  we  find 

B'  =  45°  41  '  28",            C'  =  65°  45'  58",  c  -  54°  59'  47"  ; 

whence,  by  applying  to  these  parts  the  rule  of  polar  triangles,  we 
obtain 

b—  134°  18'  32",            c=  114°  14'  2",  C=i25°o'  13". 

EXERCISES 

88.  (i.)  In  the  right-angled  spherical  triangle  ABC,  the  side  a= 
63°  56',  and  the  side  £  =  40°.  Required  the  other  side,  c,  and  the 
angles  B  and  C. 

(2.)  In  a  right-angled  triangle  ABC,  the  hypotenuse  a  =  91°  42',  and 
the  angle  ^  =  95°  6'.  Required  the  remaining  parts. 

(3.)  In  the  right-angled  triangle  ABC,  the  side  b  =  2.6°  4',  and  the 
angle  ^  =  36°.  Required  the  remaining  parts. 

-  (4.)  In  the  right-angled  spherical  triangle  ABC,  the  side  c  =  54°  30', 
and  the  angle  £  =  44°  50'.     Required  the  remaining  parts. 

Why  is  not  the  result  ambiguous  in  this  case? 


RIGHT  AND  QUADRANTAL    TRIANGLES  99 

(5.)  In  the  right-angled  spherical  triangle  ABC,  the  side  £  =  55°  28', 
and  the  side  ^  =  63°  15'.  Required  the  remaining  parts. 

(6.)  In  the  right-angled  spherical  triangle  ABC,  the  angle  B  =  69° 
20',  and  the  angle  C  =  58°  16'.  Required  the  remaining  parts. 

(7.)  In  the  spherical  triangle  ABC,  the  side  #  =  90°,  the  angle  C= 
42°  10',  and  the  angle  ^  =  115°  20'.     Required  the  remaining  parts. 
Hint. — The  angle  A  of  the  polar  triangle  is  a  right  angle. 

(8.)  In  the  spherical  triangle  ABC,  the  side  £  =  90°,  the  angle  C= 
69°  13'  46",  and  the  angle  A  =  72°  12'  4".  Required  the  remaining 
parts. 

(9.)  In  the  right-angled  spherical  triangle  ABC,  the  angle  C=23° 
27'  42",  and  the  side  b—  10°  39'  40".  Required  the  angle  B  and  the 
sides  a  and  c. 

(10.)  In  the  right  spherical  triangle  ABC,  the  angle  £  =  47°  54'  20", 
and  the  angle  C=6i°  50'  29".  Required  the  sides. 


CHAPTER   IX 
OBLIQUE-ANGLED   TRIANGLES 

89.  Let  O  be  the  centre  of  a  sphere  of  unit  radius,  and 
ABC  an  oblique-angled  spherical  triangle  formed  by  the 
three  planes  AOB,  BOC,  and  AOC.  Suppose  the  plane 


AED  passed  through  the  point  A  perpendicular  to  AO,  in- 
tersecting the  planes  A  OB,  BOC,  and  AOC,  in  AE,  ED, 
and  AD  respectively.  Then  AD=tan  b,  AE-tan  c,  OD— 


In  the  triangle  EOD, 

ED'2  =  seca£  +  secV  —  2  sec  b  sec  c  cos  a. 
In  the  triangle  AED, 

ED'*  =  tan2^  -f  tanV  —  2  tan  b  tan  c  cos  A. 
Subtracting  these  two  equations  and  remembering  that 

sec2^  —  tan2£=i,  we  have 
0  =  2  —  2  sec£  seer  cos#-|-2  tan£  tanr  cos  A. 
Reducing,  we  have 

co§c+§in& 


(i) 


OBLIQUE-ANGLED    TRIANGLES  101 

If  we  make  b  and  c  in  turn  the  base  of  the  triangle,  we  obtain  in  a 
similar  way, 

cos  £  =  cos  £•  cos#-|-sin<:  sin  a  cos  B, 
and  cos<r  =  cosd!  cos/^  +  sin<7  s\nb  cosC. 


Remark. — In  this  group  of  formulas  the  second  may  be  obtained 
from  the  first,  and  the  third  from  the  second,  by  advancing  one  letter 
in  the  cycle  as  shown  in  the  figure ;  thus,  writing  b  for 
a,  c  for  b,  a  for  c,  B  for  A,  C  for  B,  and  A  for  C.  The 
same  principle  will  apply  in  all  the  formulas  of  Oblique- 
Angled  Spherical  Triangles,  and  only  the  first  one  of 
each  group  will  be  given  in  the  text. 

90.  By  making  use  of  the  polar  triangle  where 


we  may  obtain  a  second  group  of  formulas. 

Substituting  these  values  of  a,  b,  c,  and  A  in  (i),  and  remembering 
that  cos  (  1  80°  —  A)  —  —  cos  A  and  sin  (i  So0—  A)  r=  sin  A,  we  have 

cos^4'  =  —  cos^'cosC'-fsin^'  sin  C'  cosa'. 

Since  this  is  true  for  any  triangle,  we  may  omit  the  accents  and 
write, 

cos  A  =  -  cos  B  cos  C  +  sin  B  sin  C  cos  a.  (2) 


FORMULAS   FOR   LOGARITHMIC   COMPUTATION 
.  Formula  (i),  cos  a  =  cos  b  cose  +  sin  &  sin  c  cos  A, 
cos  a  —  cos^  cos*: 


gives  cos  A  — 


sne 


By  §  36,  cos^  =  i  —  2  sin2-J^ 

cos  a  —  cos^ 


Whence      i  — 

or  sin2  £  A  — 


sin  b  sin  c 

'  sin c—cosa 


2  sin  b  sine 


102  SPHERICAL    TRIGONOMETRY 

— c]  —  cos  a 


2  sin  b  sin  c 
sin sin 


sin  b  sin  c  •          (38) 

Putting 

c  a  +  b—c  ,  a  —  b-\-c 

-=s,  then  -         — =s— c,  and  -         — =^—  b, 


we 


/sm(s 

have  sin-J^f=\/-     <—. 

V  si 


.    .    . 
sin  b  sin  c 

Since,  also,        cos  A  —  i+  2  cos*$A, 
we  have,  similarly, 

/sin  s  s\n(s  —  a] 
=  V  -      —  —  i       A 
v  - 


sin  b  sine 

/Ii 
Hence 


By  a  like  process,  formula  (2)  reduces  to 


%  ,     —cosScos(S-A)  , 

tania^W-  ^-  (TI) 


.  If,  in  formula  I,  we  advance  one  letter,  we  have 


/sin  (s  —  c)  sin  (s—a) 
=\/-  •    f       L\     • 

v       situ  sin  (s—  -b) 

And  dividing  tan^A  by  tan^^,  and  reducing,  we  obtain 

tan^A      sin(s  —  b) 

tan  \B~  sin  (j—  tf)  ' 
By  composition  and  division, 

tan  %A-\-  tan  \B      sin  (j—  ^)  +  sin(^—  a) 


tan  ^  A—  tan  ^^  ~~  sin  (j—  b)  —  sin  (j—  «)' 
§§  30»  38,  this  becomes     I?1 


§in  ^(A—  B)~~  tan  ^  (a  —  b)' 


OBLIQUE-ANGLED    TRIANGLES  103 

Multiplying  tanf  A  by  tan£#,  and  reducing,  we  obtain 


tan  \A  tan  -J  B     sin  (s—  c) 
i  sin  s 

By  division    and    composition,  and   by  §§  30,  38,  this   be- 

comes 

tanjc 


co»-^(A  —  B)      tan  -J-  (a  +  b) ' 
Proceeding  in  a  similar  way  with  formula  II,  we  obtain 

s!n-J-(a  +  6)_        cot-J-C  ,_,, 

*in%(a  —  b)~  tan%(A  —  &)' 

cow  4-  (« -f  6)  cot  -J  C 

And  ~- -.-  = ^-A — ~.  (VI) 


9»9.  In  the  spherical  triangle  yi ^{7,  suppose  C7?  drawn  per- 
pendicularly to  ABt  then,  by  the  formulas  for  right  spher- 
ical triangles, 


In  triangle  A  CD,  sin  /  =  sin  b  sin  A. 

In  triangle  BCD,  sin  p~  sin  #  sin  B. 

Whence  sin  a  sin  /?=sin  b  sin  ^4, 

sin  a     §in  6 


Remark.—  If  (A  +  B)>i8o°,  then 
1  80°,  then  (a 


,  and  if  (A-hB)< 


104 


SPHERICAL    TRIGONOMETRY 


94.  All  cases  of  oblique-angled  triangles  may  be  solved 
by  applying  one  or  more  of  the  formulas  I,  II,  III,  IV,  V, 
VI,  VII,  as  shown  in  the  following  cases. 

CASES 

(i.)  Given  three  sides,  to  find  the  angles. 

Apply  formula  I.     Check :  apply  V  or  VI. 

(2.)  Given  three  angles,  to  find  the  sides. 

Apply  formula  II.     Check  :  apply  III  or  I V. 

(3.)  Given  two  sides  and  the  included  angle. 

Apply  V  and  Vl\  and  VII.     Check :  apply  III  or  I V. 

(4.)  Given  two  angles  and  included  side. 

Apply  III  and  I V,  and  VI L     Check :  apply  V  or  VI. 

(5.)  Given  two  angles  and  an  opposite  side. 
Apply  VII,  V,  and  III.     Check :  apply  IV. 

(6.)  Given  two  sides  and  an  opposite  angle. 
Apply  VII,  V,  and  IV.     Check :  apply  III. 


EXAMPLE— CASE   (l) 
95.  Given  a  =  81°  10'  b  =  60°  20' 

To  find  A,  B,  and  C. 

a—  81°  10' 
b  —  60°  20' 

C  =:II20  25' 


<r=ii2°25' 


j  =  i26°  57'  30" 
s-a=45°  47'  30" 
^-^=66°  37'  30" 
j-^^i4°32'  30" 
log  sin  .f =9. 90259 
log  sin(.r  — ^0=9.85540 
log  sin  (s  —  £)=9. 96281 
log  sin  (j— ^=9.39982 


To  find 


sin  s  sin(s  —  <i) 
log  sin  (s  —  ^=9.9628 1 
log  sin(j-<r)=9.39982 

colog  sin  s=o.  14460 

colog  sin  (s— a)=o.ogi4i 

2) I Q .60464 

log  tan  £.4:=      9.80232 

^=32°  23'  19" 

^  =  64°  46' 38" 

ur 


OBLIQUE-ANGLED    TRIANGLES 


105 


To  find  B. 


tan  A  B=  *  /? 
V 


—  b] 


log  sin(j— a)= 

log  sin  (s— ^=9.39982 

colog  sin  .r= 0.0974 1 

colog  sin(j  —  b)  =0.03719 

2)19.38982 
logtan£#=      9.69491 


To  find  C. 


/ 
V 


_        sin  (j  —  0) 


sin  s  sin(j— c) 
log  sin  (s—a)=g. 85540 
log  sin  (s  —  ^=9.9628 1 
colog  sin  j-=o. 09741 
colog  sin  (s— r) =0.600 1 8 

2)20.51580 

log  tan£  (7=       10.25790 
K=  6l°    5'  32" 

(7=122°  II'     4" 


, 
Formula  V,  cot  A  C= 


sin  a 
A  =64°  46'  38" 
.£=  52°  42'  12" 


fl=8i°  10' 
b  =60°  20' 

=141°  30'  ;  £(a+£)=70°  45' 
a  —  b—  20°  50';  \(a  —  ^)=io°  25' 


A-B=i2°    4'  26" 
4—B)—  6°    2'  13" 

log  tan ^(A—  ^=9.02430 
log  sin  4r(rt-f-/>)=9  97501 
colog  sin  $(a  —  ^=0.74279 
cot£  (7=9.74210 
£  C-  61°    5'  32" 

(7=122°  II'     4" 


EXAMPLE— CASE  (3) 

96.  Given  a  =  78°  15'  £  =  56°  20'  C=I2O° 

To  find  ^4,  B,  and  r. 

log  sin  £(a+ ^=9.96498 
log  cos  £  (a  +  ^=9.58663 
log  si n^(«  — ^=9.27897 
log  cos  £  (a  — ^=9.99201 
log  cot  £(7=  9. 76144 


+  £)=67°  17'  30 
-3)=  10°  57'  30 


To  fin 

Formula  V/may  be  written 

cosA(r7  —  ^)  cot 


COS       (7< 
log  COS^(fl!  —  ^)=    9.99201 

log  cot  £  C=  9.  76  1  44 
colog  cos  £  («  +  b)  —  0.41337 
log  tan  $(A  +  5)  =  10.  16682 


±(A-B)=  6°  47'  4" 
A  =62°  31'  40" 
^=48°  57'  32"- 


To  find$(A-B\ 
Formula  V^  may  be  written 
sjnj>-^ 


log  sin  1(^  —  ^=9.27897 

log  cot£  (7=9.76144 

colog  sin£(rt  +^)=o.  03502 


io6 


SPHERICAL    TRIGONOMETRY 


To  find  c. 
From  Formula  VII,  sin  c= 


sin  b  sin  C 
sin  B 


log  sin£  =9.92027 

log  sin  £=9.93753' 

colog  sin  2?  =0.12249 

log  sin  ^=9. 98029 
<r=io7°8' 


Check. 
Formula  III  may  be  written 

_  sin  %  (A  +  B}  tan  £  (a  -  b} 


log  sin  %(A  +  B)  =  9.91725 

log  tan  \  (a  —  b)  =  9.28696 

colog  sin  %(A—B)  —  0.92762 

log  tan  £  r==  10.  1  3  1  83 

\c=  53°  33'  56"- 
^=107°    7'  51"  — 
(Discrepancy  due  to  omitted  decimals  ) 


AMBIGUOUS   CASES 

97.  (i.)  Two  sides  and  an  angle  opposite  one  of  them  are  the 
given  parts. 

If  the  side  opposite  the  given  angle  differs  from  po°  more  than  the 
other  given  side,  the  given  angle  and  the  side  opposite  being  either  both 
less  or  both  greater  than  90°,  there  are  two  solutions. 


(2.)  Two  angles  anc}  a  side  opposite  one  of  them  are  the  given  parts. 

If  the  angle  opposite  the  given  side  differs  from  90°  more  than  the 
other  given  angle,  the  given  side  and  the  angle  opposite  being  either 
both  less  or  both  greater  than  90°,  there  are  two  solutions. 

Remark. — There  is  no  solution  if,  in  either  of  the  formulas. 


sin  B= 


sin  A  sin  b 


sin  b  sin  A 


sin  a  sin  B 

the  numerator  of  the  fraction  is  greater  than  the  denominator. 


OBLIQUE-ANGLED    TRIANGLES 


107 


cos     /-. 


Formula  V  may  be  written 
cot  A  C-  s 


EXAMPLE — CASE   (6) 

98.  Given  #=40°  16'  £=47°44'  ^=52°  30' 

To  find  B,  B',  C,  C,  and  ct  c'. 

To  find  B  and  B'. 
Formula  VII  may  be  written 

„     sin^4  sin/' 
sm  B= : . 

sin  a 

log  sin  ^=9. 89947 

log  sin  ^=9. 86924 

colog  sin  fl=o.  1 8953 

log  sin  jB=g. 95824 

B=  65°  16'  30" 
B'  =  114°  43'  30" 

To  find  c. 
Formula  IV  may  be  written 

tan£r= 

log  CO! 

log  tan  £  («  +  />) =9. 98484 

colog  cos $(A—B)=  0.002 70 

log  tan  £  i'=g. 70080 

£<r=26°  39'  42" 
'=53°  19'  24" 
To  find  c. 

log  tan |(^  +  /')=9. 98484 

colog  cos  £  (A—  B') =0.06745 

log  tan  ££•' =9.09860 

*<•'=   7°    9'    9" 


sin£(rt  —  b) 
^)=  9.84177 

log  tan£(.4—  -5)=  9.04901  n 
colog  sin  £(rt  —  <£)=   1.1863311 
log  cot  $C=  10.0771  1 

^C=39o56'24" 
C=79°  52'  48" 

To  find  C, 

logsin£(rt  +  £)=  9.84177 
log  tai4  (.4  -.#')=  9.7815311 
colog  sin  ^  (a  —  ti\—   1.  18633  n 
log  cot  \  C  =  10.  80963 

lc=  8°  48'  41" 


Check. 
Formula  III  may  be  written 

sin  B  sin  c 
sin  b=  -  .     _     • 
smC 

log  sin  £=9.  95824 
log  sine  =9.  904  1  8 
colog  sin  C*=o.  00682 
log  sin  ^=9.86924 
^=47°  44' 


<r'  =  i4°  18'  18" 

EXERCISES 

99.  (i.)  In  the  spherical  triangle  ABC,  the  side  ^  =  124°  53',  the 
side  b  =  31°  19',  and  the  angle  A  =  16°  26'.  Find  the  other  parts. 

(2.)  In  the  oblique-angled  spherical  triangle  ABC,  angle  A  =  128° 
45',  angie  C=  30°  35',  and  the  angle  ,5  =  68°  50'.  Find  the  other  parts. 


*  The  letter  "  n"  indicates  that  these  quantities  are  negative. 


lo8  SPHERICAL    TRIGONOMETRY 

(3.)  In  the  spherical  triangle  ABC,  the  side  ^  =  78°  15',  £=56°  20', 
and  A  =  120°.  Required  the  other  parts. 

(4.)  In  the  spherical  triangle  ABC,  the  angle  ,4  =  125°  20',  the  an- 
gle (7  =  48°  30',  and  the  side  ^  =  83°  13'.  Required  the  remaining 
parts. 

(5.)  In  the  spherical  triangle  ABC,  the  side  ^  =  40°  35',  £  =  39°  10', 
and  a  =  71°  15'.  Required  the  angles. 

(6.)  In  the  spherical  triangle  ABC,  the  angle  A  =  109°  55',  B—  \  16° 
38',  and  C=  120°  43'.  Required  the  sides. 

(7.)  In  the  spherical  triangle  ABC,  the  angle  ^  =  130°  5'  22",  the 
angle  C=  36°  45'  28",  and  the  side  £  =  44°  13'  45".  Required  the  re- 
maining parts. 

(8.)  In  the  spherical  triangle  ABC,  the  angle  ^  =  33°  15'  7",  B  = 
3l0  34'  38",  and  C=  161°  25'  17".  Required  the  sides. 

(9.)  In  the  spherical  triangle  ABC,  the  side  <r=ii2°  22'  58",  £  = 
52°  39'  4",  and  a  =  89°  16'  53".  Required  the  angles. 

(10.)  In  the  spherical  triangle  ABC,  the  side  ^  =  76°  35'  36",  b  = 
50°  10'  30",  and  the  angle  ^  =  34°  15'  3".  Required  the  remaining 
parts. 

AREA  OF  THE  SPHERICAL  TRIANGLE 

100.  It  is  proved  in  geometry  that  the  area  of  a  spherical 
triangle  is  equal  to  its  spherical  excess,  that  is, 
area  =  (A  +  B  +  C—  2  rt.  angles)  X  area  of  the  tri-rectangular  triangle, 
where  A,  B,  and  C  are  the  angles  of  the  spherical  triangle. 
Hence 

area  _A+£-\-C—  180° 

surface  of  sphere  "~  720° 

The  surface  of  the  sphere  is  477^,  therefore 

A  +  B+  C-180°\ 


The  following  formula,  called  Lhuilier's  theorem,  simpli- 
fies the  derivation   of  (A  +jB+C—i8o°)  where  the  three 


OBLIQUE-ANGLED    TRIANGLES  109 

sides  of  the  spherical  triangle  are  given  ;  in  it  a,  b,  and  c 
denote  the  sides  of  the  triangle,  and  2s= 


tan  /-  _  y'tan  i  s  tan  i  (s-a)  tan  i(s-6)  tan  i  (s-cj. 


EXERCISES 

(i.)  The  angles  of  a  spherical  triangle  are,  ^=63°,  £=84°  21', 
C=79°;  the  radius  of  the  sphere  is  10  in.  What  is  the  area  of  the 
triangle  ? 

(2.)  The  sides  of  a  spherical  triangle  are,  a  =  6.47  in.,  £  =  8.39  in., 
^  =  9.43  in.;  the  radius  of  the  sphere  is  25  in.  What  is  the  area  of 
the  triangle  ? 

(3.)  In  a  spherical  triangle,  ^  =  75°  16',  ^  =  39°  20',  c  =  26  in.;  the 
radius  of  the  sphere  is  14  in.  Find  the  area  of  the  triangle. 

(4.)  In  a  spherical  triangle,  a  =  441  miles,  ^  =  287  miles,  C  =  38°  21'; 
the  radius  of  the  sphere  is  3960  miles.  Find  the  area  of  the  triangle. 


CHAPTER   X 

APPLICATIONS  TO   THE   CELESTIAL  AND  TERRES- 
TRIAL SPHERES 

ASTRONOMICAL  PROBLEMS 

101.  An  observer  at  any  place  on  the  earth's  surface 
finds  himself  seemingly  at  the  centre  of  a  sphere,  one-half 
of  which  is  the  sky  above  him.  This  sphere  is  called  the 
celestial  sphere,  and  upon  its  surface  appear  all  the  heavenly 
bodies.  The  entire  sphere  seems  to  turn  completely  around 
once  in  23  hours  and  56  minutes,  as  on  an  axis.  The  im- 
aginary axis  is  the  axis  of  the  earth  indefinitely  produced. 
The  points  in  which  it  pierces  the  celestial  sphere  appear 
stationary,  and  are  called  the  north  and  south  poles  of  the 
heavens.  The  North  Star  (Polaris)  marks  very  nearly  (with- 
in i°  16')  the  position  of  the  north  pole.  As  the  observer 
travels  towards  the  north  he  finds  that  the  north  pole  of  the 
heavens  appears  higher  and  higher  up  in  the  sky,  and  that 
its  height  above  the  horizon,  measured  in  degrees,  corre- 
sponds to  the  latitude  of  the  place  of  observation. 

The  fixed  stars  and  nebulae  preserve  the  same  relative 
positions  to  each  other.  The  sun,  moon,  planets,  and  com- 
ets change  their  positions  with  respect  to  the  fixed  stars 
continually,  the  sun  appearing  to  move  eastward  among 
the  stars  about  a  degree  a  day,  and  the  moon  about  thir- 
teen times  as  far. 


AP  PLICA  TIONS  1 1 1 

The  zenith  is  the  point  on  the  celestial  sphere  directly 
overhead. 

The  horizon  is  the  great  circle  everywhere  90°  from  the 
zenith. 

The  celestial  equator  is  the  great  circle  in  which  the 
plane  of  the  earth's  equator  if  extended  would  cut  the  ce- 
lestial sphere. 

The  ecliptic  is  the  path  on  the  celestial  sphere  described 
by  the  sun  in  its  apparent  eastward  motion  among  the  stars. 
The  ecliptic  is  a  great  circle  inclined  to  the  plane  of  the 
equator  at  an  angle  of  approximately  23^°. 

The  poles  of  the  equator  are  the  points  where  the  axis 
of  the  earth  if  produced  would  pierce  the  celestial  sphere, 
and  are  each  90°  from  the  equator. 

The  poles  of  the  ecliptic  are  each  90°  from  the  ecliptic. 

The  equinoxes  are  the  points  where  the  celestial  equa- 
tor and  ecliptic  intersect ;  that  which  the  sun  crosses  when 
coming  north  being  called  the  vernal  equinox,  and  that 
which  it  crosses  when  going  south  the  autumnal  equinox. 

The  declination  of  a  heavenly  body  is  its  distance,  meas- 
ured in  degrees,  north  or  south  of  the  celestial  equator. 

The  right  ascension  of  a  heavenly  body  is  the  distance, 
measured  in  degrees  eastward  on  the  celestial  equator,  from 
the  vernal  equinox  to  the  great  circle  passing  through  the 
poles  of  the  equator  and  this  body. 

The  celestial  latitude  of  a  heavenly  body  is  the  dis- 
tance from  the  ecliptic  measured  in  degrees  on  the  great 
circle  passing  through  the  pole  of  the  ecliptic  and  the 
body. 

The  celestial  longitude  of  a  heavenly  body  is  the  dis- 
tance, measured  in  degrees  eastward  on  the  ecliptic,  from 


112  SPHERICAL    TRIGONOMETRY 

the  vernal  equinox  to  the  great  circle  passing  through  the 
pole  of  the  ecliptic  and  the  body. 

EXERCISES 

(i.)  The  right  ascension  of  a  given  star  is  25°  35',  and  its  decima- 
tion is  -f-(north)  63°  26'.  Assuming  the  angle  between  the  celestial 
equator  and  the  ecliptic  to  be  23°  27',  find  the  celestial  latitude  and 
celestial  longitude. 


In  this  figure  AB  is  the  celestial  equator,  AC  the  ecliptic,  P  the  pole  of 
the  equator,  P'  the  pole  of  the  ecliptic.  '  S  is  the  position  of  the  star,  and 
the  lines  SB  and  SC  are  drawn  through  P  and  P'  perpendicular  to  AB  and 
AC.  AB  is  the  right  ascension  and  BS  the  declination  of  the  star,  while 
AC  is  the  longitude  and  SC  the  latitude  of  the  star. 

In  the  spherical  triangle  P'PS,  it  will  be  seen  that  P'S  is  the  comple- 
ment of  the  celestial  latitude,  PS  the  complement  of  the  declination,  and 
P'PS  is  90°  plus  the  right  ascension.  It  is  to  be  noted  that  A  is  the  ver- 
nal equinox. 

(2.)  The  declination  of  the  sun  on  December  2ist  is  — (south) 
23°  27'.  At  what  time  will  the  sun  rise  as  seen  from  a  place  whose 
latitude  is  41°  18'  north  ? 

The  arc  ZS  which  is  the  distance  from  the  zenith  to  the  centre  of  the  sun 
when  the  sun's  upper  rim  is  on  the  horizon  is  90°  50'.  The  50'  is  made  up 
of  the  sun's  semi-diameter  of  16',  plus  the  correction  for  refraction  of  34'. 


AP PLICA  TIONS  1 1 3 

(3.)  The  declination  of  the  sun  on  December  2ist  is  —  (south) 
23°  27'.  At  what  time  would  the  sun  set  as  seen  from  a  place  in  lati- 
tude 50°  35'  north  ? 


SUNRISE  SUNSET 

In  these  figures  P  is  the  pole  of  the  equator,  Z  the  zenith,  EQ  the  celes- 
tial equator.  ASh  the  declination  of  the  sun,  ZS=qcP  50',  PS—goP  +  dec- 
lination,  PZ=  90°  -latitude.  The  problem  is  to  find  the  angle  SPZ.  An 
angle  of  15°  at  the  pole  corresponds  to  I  hour  of  time. 

GEOGRAPHICAL   PROBLEMS 

102.  The  meridian  of  a  place  is  the  great  circle  passing 
through  the  place  and  the  poles  of  the  earth. 

The  latitude  of  a  place  is  the  arc  of  the  meridian  of  the 
place  extending  from  the  equator  to  the  place. 

Latitude  is  measured  north  and  south  of  the  equator  from  o°  to  90°. 

The  longitude  of  a  place  is  the  arc  of  the  equator  extend- 
ing from  the  zero  meridian  to  the  meridian  of  the  place. 
The  meridian  of  the  Greenwich  Observatory  is  usually  taken 
as  the  zero  meridian. 

Longitude  is  measured  east  or  west  from  o°  to  180°. 
The  longitude  of  a  place  is  also  the  angle  between  the  zero  meridian  and 
the  meridian  of  the  place. 


ii4  SPHERICAL    TRIGONOMETRY 

In  the  following  problems  one  minute  is  taken  equal  to  one  geo- 
graphical mile. 

(i.)  Required  the  distance  in  geographical  miles  between  two 
places,  D  and  E,  on  the  earth's  surface.  The  longitude  of  D  is  60° 
15'  E.,  and  the  latitude  20°  10'  N.  The  longitude  of  E  is  115°  20'  E., 
and  the  latitude  37°  20'  N. 


In  this  figure  A  C  represents  the  equator  of  the  earth,  P  the  north  pole, 
and  A  the  intersection  of  the  meridian  of  Greenwich  with  the  equator.  PB 
and  PC  represent  meridians  drawn  through  D  and  E  respectively.  Then 
AB  is  the  longitude  and  BD  the  latitude  of  D  ;  AC  the  longitude  and  CE 
the  latitude  of  E. 

(2.)  Required  the  distance  from  New  York,  latitude  40°  43'  N., 
longitude  74°  o'  W.,  to  San  Francisco,  latitude  37°  48'  N.,  longitude 
122°  28'  W.,  on  the  shortest  route. 

(3.)  Required  the  distance  from  Sandy  Hook,  latitude  40°  28'  N., 
longitude  74°  i'  W.,  to  Madeira,  in  latitude  32°  28'  N.,  longitude  16°  55, 
W.,  on  the  shortest  route. 

(4.)  Required  the  distance  from  San  Francisco,  latitude  37°  48' 
N.,  longitude  122°  28'  W.,  to  Batavia  in  Java,  latitude  6°  9'  S.,  longi- 
tude 1 06°  53'  E.,  on  the  shortest  route. 

(5.)  Required  the  distance  from  San  Francisco,  latitude  37°  48' 
N.,  longitude  122°  28'  W.,  to  Valparaiso,  latitude  33°  2'  S.,  longitude 
71°  41'  W.,  on  the  shortest  route. 


CHAPTER   XI 

GRAPHICAL  SOLUTION   OF  A   SPHERICAL  TRIANGLE 

J.03.  The  given  parts  of  a  spherical  triangle  may  be  laid 
off,  and  then  the  required  parts  may  be  measured,  by  making 
use  of  a  globe  fitted  to  a  hemispherical  cup. 

The  sides  of  the  spherical  triangle  are  arcs  of  great  circles, 
and  may  be  drawn  on  the  globe  with  a  pencil,  using  the 
rim  of  the  cup,  which  is  a  great  circle,  as  a  ruler.  The  rim 
of  the  cup  is  graduated  from  o°  to  1 80°  in  both  directions. 

The  angle  of  a  spherical  triangle  may  be  measured  on  a 
great  circle  drawn  on  the  sphere  at  a  distance  of  90°  from 
the  vertex  of  the  angle.* 

CASE  I.  Given  the  sides  a,  b,  and  c  of  a  spherical  triangle, 
to  determine  the  angles  A ,  B,  and  C. 

Place  the  globe  in  the  cup,  and  draw  upon  it  a  line  equal 
to  the  number  of  degrees  in  the  side  c,  using  the  rim  of  the 
cup  as  a  ruler.  Mark  the  extremities  of  this  line  A  and  B. 
With  A  and  B  as  centres,  and  b  and  a  respectively  as  radii, 
draw  with  the  dividers  two  arcs  intersecting  at  C  (Fig.  i). 
Then,  placing  the  globe  in  the  cup  so  that  the  points  A  and 
C  shall  rest  on  the  rim,  draw  the  line  AC=b,  and  in  the 
same  way  draw  BC=a. 

To  measure  the  angle  A  place  the  arc  AB  in  coincidence 

*  Slated  globes,  three  inches  in  diameter,  made  of  papier-mache,  and  held 
in  metal  hemispherical  cups,  are  manufactured  for  the  use  of  students  of 
spherical  trigonometry  at  a  small  cost. 


Ii6  SPHERICAL  TRIGONOMETRY 

with  the  rim  of  the  cup,  and  make  AE  equal  to  90°.  Also 
make  AF  in  AC  produced  equal  to  90°.  Then  place  the 
globe  in  the  cup  so  that  E  and  F  shall  be  in  the  rim,  and 
note  the  measure  of  the  arc  EF.  This  is  the  measure  of  the 
angle  A.  In  the  same  way  the  angles  B  and  C  can  be  de- 
termined. 


CASE  II.  Given  the  angles  A,  B,  and  C,  to  find  the  sides 
a,  b,  and  c. 

Subtract  A,  B,  and  C  each  from  180°,  to  obtain  the  sides 
a1 ',  b' ,  and  c'  of  the  polar  triangle.  Construct  this  polar  tri- 
angle according  to  the  method  employed  in  Case  I.  Mark 
its  vertices  A',  B' ,  and  C '.  With  each  of  these  vertices  as 
a  centre,  and  a  radius  equal  to  90°,  describe  arcs  with  the  di- 
viders. The  points  of  intersection  of  these  arcs  will  be  the 
vertices  A,  £,  and  C  of  the  given  triangle.  The  sides  of 
this  triangle  a,  b,  and  c  can  then  be  measured  on  the  rim 
of  the  cup. 


GRAPHICAL  SOLUTION 


117 


CASE  III.  Given  two  sides,  b  and  c,  and  the  included  angle 
A,  to  find  B,  C,  and  a. 

Lay  off  (Fig.  3)  the  line  AB  equal  to  c,  and  mark  the 
point  D  in  AB  produced,  so  that  AD  equals  90°.  With  the 
dividers  mark  another  point,  F3  at  a  distance  of  90°  from  A. 
Turn  the  globe  in  the  cup  till  D  and  Fare  both  in  the  rim, 
and  make  DE  equal  to  the  number  of  degrees  in  the  angle  A. 
With  A  and  E  in  the  rim  of  the  cup,  draw  the  line  AC  equal 
to'  the  number  of  degrees  in  the  side  b.  Join  C  and  B.  The 
required  parts  of  the  triangle  can  then  be  measured. 


FIG.  3 


FIG.  4 


CASE  IV.  Given  the  angles  A  and  B  and  the  included  side 
c,  to  find  a,  b,  and  C. 

Lay  off  the  line  AB  equal  to  c.  Then  construct  the  given 
angles  at  A  and  B,  as  in  Case  III.,  and  extend  their  sides  to 
intersect  at  C. 

CASE  V.  Given  the  sides  b,  a,  and  the  angle  A  opposite  one 
of  these  sides,  to  find  c,  B,  and  C.  (Ambiguous  case.) 


Ii8  SPHERICAL    TRIGONOMETRY 

Lay  off  (Fig.  4)  AC  equal  to  b,  and  construct  the  angle  A 
as  in  Case  III.  Take  c  in  the  dividers  as  a  radius,  and  with 
C  as  a  centre  describe  arcs  cutting  the  other  side  of  the  tri- 
angle in  B  and  B' ,  and  measure  the  remaining  parts  of  the 
two  triangles. 

If  the  arc  described  with  C  as  a  centre  does  not  cut  the  other  side  of  the 
triangle,  there  is  no  solution.     If  tangent,  there  is  one  solution. 

CASE  VI.  Given  the  angles  A,  B,  and  the  side  a  opposite 
one  of  the  angles. 

Construct  the  polar  triangle  of  the  given  triangle  by 
Case  V. ;  then  construct  the  original  triangle  as  in  Case  II., 
and  measure  the  parts  required. 

The  constructions  given  above  include  all  cases  of  right  and  quadrantal 
triangles. 


CHAPTER   XII 
RECAPITULATION    OF    FORMULAS 

ELEMENTARY    RELATIONS   (§   IO) 

sin  x  *  COSJT 

tan  x  =  -  ,  cot  x  =  —  -  , 

COSJT  sinx 

i  i 

sec  x  =  --  ,  esc  x  = 

cos.r 

tan  x  cot  x  =  i  , 
sin3  x  -\-  cos2  x=.  i, 


i  +  cot2  x  =  csca  x. 
RIGHT    TRIANGLES   (§§  14  AND   2  7) 


cos  A  =  -  ,  cos  B  =  -  , 

|,  tan  #  =  -, 


cot  A  =  -  ,  cot  B  =  -,  , 

a  b 


where  c=.  hypotenuse,  a  and  b  sides  about  the  right  angle;  A  and  B 
the  acute  angles  opposite  a  and  £. 

FUNCTIONS   OF  TWO   ANGLES   (§§  30-34) 

sin  (x-\-y)=.s\nx  cos^-j-cos^r  sinj, 
sin  (x—  y)  =  sinx  cos  y  —  cos  x  siny, 
cos  (x  -\-y)  =  cos  x  cosy  —  sin  x  siny, 
cos  (.r—  j)  =  cos.r  cosj-j-sin^r  si 


120  RECAPITULATION  OF  FORMULAS 


tan  (.r-h)/)=- 

i  —  tan^r 

tan  x  —  tan  y 

tan  (.r—  y)  =  —  —  — 

i-f  tan^r  tanj/ 

cot^r  cot/—  i 


cot^r  cot  y+  i 
cot  (.r  — /)=  —  —  . 

cot/  —  cot  x 

FUNCTIONS   OF  TWICE  AN  ANGLE  (§  36) 
sin  2x  =  2  sin^r  cos,r, 


=  2  cos2.r  —  i, 
2  tan  .r 


tan  2x  = 
cot  2;r  = 


i  — tan  x 
cot'.r— i 


2  cot  x 
FUNCTIONS   OF   HALF  AN   ANGLE  (§  37) 


cos 


i  —  cos  x 


SUMS  AND   DIFFERENCES   OF  FUNCTIONS  (§  38) 

sin  //  -f~  sin  ^  =  2  sin  ^  (« -+-  v)  cos  ^  («  —  T/), 
sin  u  —  sin7/  =  2  cos^(«  +  z/)  sin  \(u — v\ 

COS  U  -f-  COS  W  =  2  COS  £  («  +  7/)  COS  £  («  —  2/), 

cos  u  —  cos  ?/  =  —  2  sin  $  (w  +  v)  sin  £  (w  —  v). 
sin  u -f  sin  y  _  tan  \  (u  -f  ^) 
sin  u  —  sin  v ~~  tan  \  (u  —  v] ' 


RECAPITULATION  OF  FORMULAS  121 

OBLIQUE   TRIANGLES  (§§  42-45) 

a_sin  A  a__s'mA  b      sin  B 

6~sinL"  £~s'mC'  <r~sinC' 

a  —  ^tan|(^~  £) 


tan£C'  = 

'.'—*) 

a+fi+c    ' 
where  s= 


where  AT= 


AREA   OF   A   TRIANGLE  (§  46) 

=|rtr  sin  ^.      5"=^  sin  C      5=^  sin  ^4. 


LOGARITHMIC,  COSINE,   SINE,   AND   EXPONENTIAL  SERIES 

(§58) 


=*-   +~+>  etc> 


122  RECAPITULATION  OF  FORMULAS 


,. 
r2      r3      x* 

^=i+.r+-  +  _+-+,etc. 

DE    MOIVRE'S   THEOREM   (§  71) 


-     ~  *  -  cosw~3  ^  sin3^+,  etc. 

(«  —  i  ) 
cos  «.r  rr  >z  cos  .r  --  -  —  ?  —  -  CO8*      jr  sin2^-f,  etc. 


HYPERBOLIC   FUNCTIONS   (§  75) 

ex-e~x 


. 
cos  /.r  =  —  =cosh  jr. 

2 


SPHERICAL  TRIANGLES 
RIGHT   AND    QUADRANTAL   TRIANGLES  (§§   83,  87) 

Use  Napier's  rules. 

OBLIQUE   TRIANGLES    (§§  89-93) 

cos  a  =.  cos  b  cos  c  -j-  sin  b  sin  c  cos  ^4. 
cos  A  =  —  cos  #  cos  C-\-  sin  ^  sin  C  cos  #. 


sin  s  sin  (.$•  —  a) 


RECAPITULATION  OF  FORMULAS  123 

•v; 


tan  |  a  -cos      cos      - 


cos  (S—£)  cos(S-C) 
tan     c 


sn      ^   — 


cos     (y  —    )     tan 
sin  $(a-\-&)  cot  ^ 


sini(«  —  b)     tani(A  — 
cos$(a-\-t>)_      cotj  C 


cos  £  (#  —  ^)~"tan  £  (A-\-£)' 
s\na  __  sin  If 
sin  A  !  ~sin  B' 

AREA  OF  SPHERICAL  TRIANGLES   (§  lOl) 


tan  (*'  — 

\  4 


APPENDIX 

RELATIONS   OF  THE   PLANE,  SPHERICAL,  AND   PSEUDO- 
SPHERICAL  TRIGONOMETRIES 

We  have  up  to  the  present  considered  the  trigonometries 
which  deal  with  figures  on  a  plane  or  spherical  surface.  A 
characteristic  feature  of  these  two  surfaces  is  that  the  curv- 
ature of  the  plane  is  zero,  while  that  of  the  sphere  is  a  posi- 
tive constant  p.  If  the  radius  of  the  sphere  is  increased  in- 
definitely, its  surface  approaches  the  plane  as  a  limit  while 
its  curvature  p  approaches  o. 

-In  works  on  absolute  geometry  it  is  shown  that  there  ex- 
ists a  surface  which  has  a  constant  negative  curvature :  it  is 
called  a  pseudo-sphere,  and  the  trigonometry  upon  it  pseudo- 
spherical  trigonometry. 

We  observe  that  as  p  passes  continuously  from  positive 
to  negative  values,  we  pass  from  the  sphere  through  the 
plane  to  the  pseudo-sphere.  Thus  the  formulas  of  plane 
trigonometry  are  the  limiting  cases  of  those  of  either  of  the 
two  other  trigonometries. 

In  the  treatment  of  spherical  trigonometry  the  radius  of 
the  sphere  has  been  taken  as  unity.  If,  however,  the  radius 
of  the  sphere  is  r,  and  a,  b,  and  c  denote  the  lengths  of  the 
sides  of  the  spherical  triangle,  the  formulas  are  changed,  in 

that  a  is  replaced  by  -,   b  by  -,   and  c  by  -  ;  thus, 


126 


APPENDIX 


becomes 


.     „     sin^r 
sin  C=  — 

sin  a 


.    c 

sin- 

r 

.    a 

sm- 

r 


The  formulas  for  pseudo-spherical  trigonometry  are  the 
same  as  the  formulas  of  spherical  trigonometry,  except  that 

the  hyperbolic  functions  of  -,    -,  and  -  are  substituted  for 

the  trigonometric. 

Thus,  corresponding  to  the  above  formula  of  spherical 
trigonometry,  is  the  formula 


sin  C  = 


of  pseudo-spherical  trigonometry. 


The  pseudo-sphere  is  generated  by  revolving  the  curve  whose  equation  is 
r-\-  vr't  —  x1 


y=r  log 

*A> 

about  its  y  axis.     The  radius  of  the  base  of  the  pseudo-sphere  is  r. 


APPENDIX  127 

Hence  the  formulas  of  plane  trigonometry  can  be  derived 
from  the  formulas  of  either  spherical  or  pseudo-  spherical 
trigonometry  by  expressing  the  functions  in  series  and  al- 
lowing r  to  increase  without  limit. 

Example.—  Show  that  if  r  is  increased  indefinitely  the  following 
corresponding  formulas  for  the  spherical  and  pseudo-spherical  right 

triangle 

a  be 

cos    =  cos  -  cos  -  »  (i) 

,  a  i  b       1  c 

cosh  -  =  cosh-  cosh-,  (2) 

r  r  r 

reduce  to  the  corresponding  formula  for  a  plane  right  triangle;  that 
is,  to 

a>=F+c\  (3) 

Substituting  the  series  cos  -,  etc.,  in  equation  (i),  we  obtain 


I    rt!  ,    1    a4  ,  I    If       I     S       i    A4 

'-i!  7<  +4-,  S  +  '  '  •  =  '-  T\  ?"  7  +  r,  ?+  '  '  '         ^ 

Substituting  in  equation  (2)  the  series  for  cosh  -  ,  etc.  ,  which  we  obtain  from 


cosh  x  =  -  ,  we  have 


Cancelling  i  in  equations  (4)  and  (5),  multiplying  by  r2,  and,  finally,  allowing 
r  to  increase  without  limit,  we  get  from  either  equation 


EXERCISES 

Derive  each  of  the  following  formulas  of  plane  trigonometry  from 
the  corresponding  formula  of  spherical  trigonometry,  and  also  from 
the  corresponding  formula  of  pseudo-spherical  trigonometry. 


123  APPENDIX 

Right  triangles  ;  A  =  right  angle. 
(i.)  Plane,  sin  C=^ 

sin  c 

Spherical,  sin  C  =  - 

sin  a 

Pseudo-spherical,        sin  C=  .  h   ' 

Oblique  Triangles. 
(2.)  Plane,  # 2  =£2  +  <r2  —  2  be  cos  A 

Spherical,  cos  a  =  cos  £  cos  c-\-  sin  £  sin  <r  cos  A 

Pseudo-spherical,  cosh  #  =  cosh  b  cosh  <:  -J-  sinh  b  sinh  ^  cos  A. 


(3.)  Plane,  S=Vs  (s-a)  (j-  J)  (j-f). 

Spherical, 
(^  +  3+  C-  .800)  =     i      ^  £  tan   (J-.^  ^  ^a-g  ^  ^^-^ 

4  *  r  r  y  y 

Pseudo-spherical, 


(s-a)  ,  (s-6) 

i_2  tanh  ^  1—  -i  tanh  |  L- 


§  4  (page  3). 

(i.)  192°  51'  25f . 

Quadrant  III. 

(2.)    25°. 

(3-)  287°,  647^. 
(4.)  Quadrant  III. 

§  9  (page  9). 

tan  1000°  is  negative, 
cos  8 10°  is  o. 
sin  760°  is  positive, 
cot  —  70°  is  negative, 
cos  —  550°  is  negative, 
tan  —  560°  is  negative, 
sec  300°  is  positive, 
cot  1560°  is  negative, 
sin  130°  is  positive, 
cos  260°  is  negative, 
tan  310°  is  negative. 

§  13  (page  n). 
(3.)  cos  —  30°  =  ^  v/3- 
tan  -3o°  =  -iy'3. 

COt  —  30°  =  —  v/3, 

sec-3op=§yg; 

CSC  —30°=  — 2. 

(4.)  cos.r=  -§  v/2, 
tan  x  =  i  v/2, 

COt  X  =  2  V/2, 

sec.r  =  — I  v/2, 
esc  .r  =  —  3. 
9 


ANSWERS   TO    EXERCISES 

(5-) 


cot  y  =  —  §,    sec  y  =  J, 


(6.)  sin  60°  =  £  v/3. 

tan  60°  =  v/3, 

cot  60°  =  i  ^3. 

sec  60°  =  2, 

esc  60°  =  f  v/3. 
(7-)  cos  o°  =  i  ,    tan  o°  =  o. 
(8.)  sin  2  =  |,    cos  ,3-  =  !, 


esc  ^  =  f  . 

(9.)  sin  45°  =  cos  45°  =  |  -/2, 
tan  45°=  i, 

sec  45°  =  esc  45°  =  v/2. 
(10.)  sin^  =  —  £  v/5,  cosj/  =  —  f, 
cot_>/  =  f  v/5,    sec  j  =  —  |, 


(11.)  sin3o°  =  i  co 
tan  30°  =  i-  v/3, 
sec  30°  =  |  v/3, 

CSC  30°  =  2. 

(12.) 


=—  f. 


§  17  (page  14). 

(i  .)  sin  70°  =  cos  20°, 
cos  60°  =  sin  30°, 
cos  89°  31'=  sin  29', 
cot  47°=  tan  43°, 


'3° 


ANSWERS   TO  EXERCISES 


tan  63°=  cot  27°, 
sin  72°  39'=  cos  17°  21'. 
(2.)  .r  =  300. 

(3.)  *  =  22°30'. 

(4.).r=i8°. 
(5.)  jr=is°. 

§  25  (page  21). 

(i.)  225°  and  315°, 

60°  and  240°. 
(2.)  60°,  120°,  420°,  480°. 
(3.)  sin-3o°=-i 

cos  — 30°=^  -v/3, 

sin  765°=  cos  765  =  £  -v/2, 

sin  1 20°=  £ -v/3, 

cos  1 20°  =  —  |, 

sin  210°=  —  ^, 

cos  2io°=— £  -y/3- 

(4.)   The   functions   of  405°  are 

equal  to  the  functions  of  45°. 

sin  6oo°=  —  |-  \/3» 
cos6oo°=— i 
tan  600°  =  -Y/3. 
cot  6oo°=£  -v/3, 
sec  6oo°=  —2, 
esc  6oo°=  — f  -v/3. 

The  functions  of  1125°  are 
equal  to  the  functions  of  45° 
sin  —  45°  =  —  ItV*. 
cos- 45°=  i  -v/2, 
tan  —  45°=  cot  —  45°=  —  i , 
sec  —  45°=-v/2, 
csc  — 45°=  — -v/2. 
sin  225°=  cos  225°=  —  £  V2 
tan  225°=  cot.  225°=  i, 
sec  225°=  esc  225°=  —  v/2. 
(5.)  The  functions  of  —  120°  are 


the  same  as  those  of  600° 

given  in  (4). 

sin  —  225°  =  £  1/2, 

cos  —  225°  =  —  £  -v/2, 

tan  —  225°=  cot  —  225°=  —  i, 

sec  — 225°=  —  "v/2, 

esc  —  225°= -v/2, 

sin  —  420°  =  —  £  \/3, 

cos  — 420°  =\, 

tan  —  420°  =  —  y^ 

cot— 420°=— iVi 

sec  —  420°  =  2, 

csc-42o°  =  -t-v/3^ 

The  functions  of  3270°  are 

equal  to  the  functions  of  30°. 
(6.)  sin  233°  =  — cos  37°, 

cos  233°  —  —  sin  37°, 

tan  233°  —  cot  37°, 

cot  233°  =  tan  37°, 

sec  233°  =  —  esc  37°, 

esc  233°  =  — sec  37°. 

sin  —  1 97°  =  sin  17°, 

cos  —  1 97°  =  —  cos  1 7°, 

tan  — 197°  =  — tan  17°, 

cot— 197°  =  — cot  17°, 

sec  —  1 97°  =  —  sec  17°, 

esc—  1 97°  =  esc  17°. 

sin  894°  =  sin  6°, 

cos  894°  =  —  cos  6°, 

tan  894°  =  —  tan  6°, 

cot  894°  =  —  cot  6°, 

sec  894°  =  —  sec  6°, 

esc  894°  =  esc  6°. 
(7.)  sin  267°=:— sin  87°, 

tan  —  254°  =  —  tan  74°, 

cos  950°  =  —  cos  50°. 
(8.)  —0.28. 


ANSWERS   TO  EXERCISES 


(9.)  2  sin2  x. 

(10.)  i  -f-seca  x. 

(ii.)  sin  (*— 90°)=  —  cos. r, 
cos(.r  —  ooc)  =  sin-r, 
tan  (JT  —  90°)  =  —  cot  x, 
cot  (JT  —  90°)  =  —  tan  x, 
sec  (x  —  90°)  =  esc  x, 
esc  (.r  —  90°)  =  —  sec  x. 

$  28  (page  24). 

(I.)    a  =62.324, 

^  =  32°  52'  40". 
(2.)      £  =  21.874, 

^  =  39°  45'  28", 
#=50°  14'  32". 
(3.)    <*  =  300.95. 
£  =  683.96, 
£  =  66°  15'. 
(4.)        £  =  26.608, 
*•  =  45-763. 
^  =  35°  33'- 
area  =  495. 34. 

(5.)        £  =  3-9973- 
?  =  4.1537, 

^  =  1 5°  46'  33". 
area  =  2. 257. 
(6.)  £  =  0.01729. 
(7.)  <*  =  298.5. 
(8.)  ^  =  39°  42' 24". 
(9.)  ^-  =  2346.7. 
(10.)  #  =  28°  57' 8". 
(if.)  444.16  ft. 

(12.)    186.32  ft. 

(i 3.)  34°  33' 44". 

114.)  303.99  ft. 

(15.)  238.33  ft. 

(16.)  15  miles  (about). 

(17.)  79,079  ft. 

(18.)  165.68  ft. 


(I9-)  53°  33'- 
(20.)  115.136  ft. 

(21.)    76.355   ft. 
(22.)   £  =  80°  32", 

A  =  C  =  49°  59' 44". 
(23.)      #=53°i6'36", 
£=  12.0518  in., 

area  =  72. 392  sq.  in. 
(24.)        £=  130.52  in., 

area  =  24246  sq.  in. 
(25.)  23.263  ft. 
(26.)  1 7°  48". 
(27.)  5.3546  in. 
(28.)  1084950  sq.  ft. 
(29.)  17  ft.,  885  sq.  ft. 
(30.)       radius  =  24.882  in., 

apothem  =  20. 1 3  in., 

area=  1472  sq.  in. 
(31.)  12.861. 
(32.)  1782.3  sq.  ft. 
(33.)  38168  ft. 

(34.)    20.21   ft. 

(35.)  2518.2  ft. 

§  29  (page  28). 

(I.)   ^  =  22°  58', 
£  =  7.07, 

c  =  9.0046. 
(2.)    £  =  79-435. 
A  =  45°  27'  14", 
C  =  95°  24'  46". 
(3.)      ^^  =  7.6745, 
^#'  =  2.6435, 
^  =  46°  43'  50", 
2? '  =  133°  16'  10", 
ACB^  105°  53'  10", 
ACB'  =  19°  20'  50". 

(4.)  ^  =  37°  53'- 
#  =  43°  52' 25", 


132 


ANSWERS  TO  EXERCISES 


C  =  98-14'  35"- 
(5.)  902.94. 
(6.)  1253.2  ft. 
(7.)  357-224  ft. 

(8.)       ^  =  44°  2'  9". 
^  =  51°  28'  ii", 
C  =  84°  29'  40", 
area  =  126100  sq.  ft. 
(9.)  407.89  ft. 
(io.)  B=i2\°  7'  16", 
C  =  92°  20'  38", 
D  =  J\°  n'  6". 
(11.)  #^  =  6.6885, 


v      3V/5  +  : 
cos  (*  —y)  —  2JL5 — 


§  39  (Page  37). 
(5.)  sin  (45°-*)  = 

£  1/2  (cos  *  —  sin  *), 
cos  (45°—*)=: 

1 1/2  (cos *  + sin*), 
sin  (45°+*)  = 

1/2  (cos*  +  sin*), 


§  34  (page  34). 

(2.)  sin  (45°+  x)  = 

£l/2  (cos  *  +  sin*), 
cos  (45°+  f)  = 

4  I/2  (cos  .r  —  sin*), 
sin  (30°—*)  = 

£  (cos*—  1/3  sin*), 
cos  (30°—  *)_  = 

i  (VX3  cos  *  +  sin*),     | 
sin  (6o°+*)_= 

$  (y'3  cos*  -f-  sin*), 
cos  (60°+*)  = 

£  (cos*  —  -y/3  sin*). 
(3.)  sin(*+y)=ff, 
sin  *—   )  =     - 


(4.)  sin  75°  = 


cos  7  5°  = 


(.5-)  sin  15°= 


4 
1/6+  1/2" 


(15.)  sin  2*  =  —  ff, 
cos  2*  =  —  T/5. 

(i  6.)  sin  22|°  =         2  -1/2. 


(170 


cos  i5   = 


ANSWERS    TO  EXERCISES 


133 


tan  i5°  =  2  —  v/3, 
cot  15°  =  2-f-  V^. 
sec  1 5°  =  2^/2  —  \/3, 

CSC  I5°  =  2*/2  +  -v/3. 

(20.)   sin  5.r  = 

5  sin  .r  — 20  sin3  .r 

+  1 6  sin5.r. 
(21.)  cos  5_r  = 

5  cos  x  —  20  cos3  x 

4-  1 6  cos5  ,r. 

(23.)  The  values  of  .r  <3oo°  are 

o°,  30°,  1 50°,  1 80°,  210°,  330°. 

(36.)  tan,v  tan^/. 

§  41  (page  40). 

(i.)  sin-«i  ^2=45°.  i 

45°+ 360°,  etc., 
cos- i  £  =  60°,  300°,  etc., 
tan-'  (—0=  1 35°. 3 1 5°. etc., 
cos  —  1  i  =0°,  360°,  etc., 
sin -i  (—  |)  =  210°,  330°,  etc. 

(2.)  tan,r  =  3- 

(3.)  cos.r  =  d 

(5.)  sin  (cos-1  $)  =  ±jj. 
(6.)  cot  (tan- 1  3V)  =17. 

(7.)  «  =  i\/3. 

(8.)  45°,  225°. 

(9.)  ,r  =  45°,_y  =180°. 

(10.)  sin  —  Irt  =  225°. 

§  48  (page  46). 

(i.)  C=i2i°33', 

^  =  2133.5, 

c  =  2477.8. 
(2.)  C=55°4i', 

^  =  534.05, 


^-  =  653.52. 

(3.)  C=45°34/, 

a=  1548.1, 

(4.)  ^  =  105°  59', 
a  =54.018, 

^•  =  47.738. 
(5.)  ^  =  68°  58', 
^  =  5274.9, 
£•  =  3730. 

(6.)  ^  =  54°  58'. 
a  =  923.4, 

c=  1 187.7- 

§  49  (page  47). 

(i.)  (I.)  Two  solutions. 

(2.)  One  solution,  a  right  tri- 
angle. 

(3.)  One  solution.. 

(4.)  Two  solutions. 
(2.)  Z?=i6°57'2i", 

C=  1 5°  50'  39"'. 

£•=1:0.32122. 
(3.)    £-  =  2.5719, 

B=IT°  15'  i", 

C=i42°  1 3' 59". 
(4.)  £-  =  93-59.         c'  =  54-069, 
B  =  26°  52'  7",  B'=  1 33°  7'  53", 
C  =  i3i°46/53",C/=25°3i/7'/. 
(5.)  No  solution. 
(6.)  £=  1.0916,        ^'=0.36276, 

B  =  1 1 7°  50'  44",  B'=  1 7°  5'  1 6". 

§  50  (page  48). 

(i.)    a  =  0.0971, 
#=90°  35' 36", 

5  =  0.0053261. 


134 


ANSIVERS    TO   EXERCISES 


(2.)     C  —  14.211, 

^  =  48°  44'  32", 

A  =  76°  20'  5", 

C=  95°  1  5'  56", 

£  =  44°  52'  55" 

5  =  0.60709. 

5  =  80.962. 

(3.)    £  =  85.892, 

§  52  (page  50). 

A  =  67°  2l'42", 

(i.)  1116.6  ft. 

C  =  62°48'  18", 
5=3962.8. 
(4.)    «  =  0.6767, 

(2.)  308  1.  8  yards. 
(3-)  638.34  ft., 
14653  sq.  ft. 

5=  15°  9'  2l", 

(4.)  4.1  and  8.1. 

C=  131°  19'  39", 

(5.)  13.27  miles. 

5=0.08141. 

(6.)  6667  ft.     One  solution. 

(5.)    <:  =  72.87, 

(7.)  121.97. 

^  =  40°  50'  32", 

(8.)  44°  2'  56". 

^=11°  2'  28", 
5  =  422.65. 

(9.)  32.151  sq.  miles. 
(11.)  54°  29'  12". 

(12.)  a  =  12296  ft., 

§  51  (page  49). 

r=  13055  ft. 

(i.)  ,4  =  55°  20'  42", 
^=106°  35'  36", 
C=i8°3'42", 
5=267.92. 
(2.)  A  =  34°  24'  26", 
B  =  73°  H'  56", 

C=72°20'  36", 

5=3.6143. 

(13.)  294.77  ft. 

(14.)   222.1   ft. 
(16.)  42Q2J^ftr    4-  ' 

(17.)  72.613  miles, 
(i  8.)  50.977  ft. 
(19.)  0.85872  miles. 
(20.)  2.98  miles. 
(21.)  1393.9  ^. 
(22.)  8.2  miles. 

(3.)    /*  =  52°  20'  24", 

B=  107°  19'  14", 

(23.)  187.39  ft. 
(24.)  0.6011. 

C  =  20°  20'  24", 

(25.)  4.8112  miles. 

k 

5=  1437.5. 

(4.)  A  =97°  48', 

(26.)  60°  51'  8". 

#=l8°2I    48", 

(27.)  37.365  ft. 

C=63°  50'  12", 

(28.)  3.2103  miles. 

5=193.13. 

(29.)   10.532  miles. 

(5.)  A  =  54°  20'  16", 

(30.)  851.22  yards. 

B  =  70°  27'  46", 

(31.)  9.5722  miles. 

C=54°72', 

(32.)  6.1271  miles. 

5  =  6090. 

(33.)  280.47  ft. 

(6.)  A  =  35°  59'  30", 

(34-)  123.33  ft. 

ANSWERS    TO  EXERCISES 


135 


(35-)  4-8ii2  miles. 
(36.)  2666.1  ft. 

*  53  (page  56). 

(i.)  30°  =  0.5236, 
45°  =  0.7854, 
60°  =  1.0472, 

I  20  =  2.0944, 
135°=  2.3562, 
720°=  12.5664, 
990°  =17.2788. 

<2.)    I  =  22°  30', 
1O 

|  =  28^  38'  53", 
}  =  100°  16'  4". 
(3.)  1.35,0.54. 

§  74  (page  73). 

(i.)  sin  4,1-  =  4  cos3. i~  sin  x 

—  4  cos.r  sin3^-, 
cos  4,1-  =  cos4  x 

—  6  cos2  x  sin2  x -j-  sin4  x. 
(2.)  sin  6x  =  6  cos5  .r  sin  .r 

—  20  cos3 a-  sin3.r 
-|-6  cos.r  sin5^-, 
cos  6x  =  cos6  .r 

—  15  cos*.r  sin2^- 
-|-  1 5  cos1'  x  sin4  x  —  sin6  x. 

1 1   \      i-  v  1      I      „•    V    3 

(3-)-ro— '•    -ri  =  \  -r z  —  > 


(4.)  .r0=i,  ^  =  o.  3090+  /  0.951  1, 
-fa  =  —  0.8090  -|-  /'  0.5878, 
.r3  =  —  0.8090  —  /  0.5878. 
.r4  =  o.  3090  —  /  0.951  1. 

§  77  (page  78). 

(23.)  .r  =  3o°. 
(24.)  ^  =  30°. 
(25.)  x  =  o°  or  45°. 
(26.)  A-  =  6o°. 
(27.)_y  =  45°. 
(28.)  j  =  45°- 
(29.)  -r  =  45°. 
(30.)  .r  =  3o°. 
(31.)  ^-  =  60°. 
(32.)  ^-  =  30°. 
(33.)  No  angle  <  90°. 
(34.)  jr  =  3o-. 
(35.)  sin  92°  =  cos  2°. 
(36.)  cos  127°  =  —  sin  37°. 
(37.)  tan  320°  =  —  tan  40°. 
(38.)  cot  350°  =  —  cot  ioj. 
(39.)  sin  265°  =  —  cos  5°. 
(40.)  tan  171°=  -tan  9°. 
(41.)  cos.r=  - 
tan.  r  ^  — 


esc  -r  — 


(42.) 


(43.)  sin.r  =  — 
cos  x-~ 
cot  .1-  =  f  ,  sec  .r  =  — 


136 


ANSWERS    TO  EXERCISES 


(44.)  sin,r  =  —  7^-  -v/74. 


=  —  f,  sec  jt-  = 


(45.)  Quadrant  II  or  IV. 
(46.)  Quadrant  I  or  II. 
(47.)  Quadrant  III  or  IV. 
(48.)  Quadrant  I  or  II. 
(49.)  .r  =  o°,  120°,  1  80°,  240°. 
(50.)  .r  =  3o°,  135°,  150°,  315°. 
(51.)  ,r  =  o°,  90°,  120°,  1  80°,  240 

270°. 
(57-)  o. 
(58.)  a. 
(59.)  2  (<*—£). 
(60.)  i(fl'-J9). 

§  78  (page  80). 

(i.)  306.32  ft. 

(2.)    831.06  ft. 

(3.)  53°  28'  14". 
(4.)  49.39  ft. 
(5.)  0.43498  mile. 
(6.)  209.53  ft. 
(7.)  7-3188  ft- 
(8.)  37°  36'  30". 
(9.)  109,28  ft. 
(10.)  502.46  ft. 
(u.)  6799.8ft. 

(12.)  219.05  ft. 

(13.)  49i.76ft. 

(14.)  50°  32'  44". 

(15.)  49°  44'  38". 

(i  6.)  34-063  ft. 

(17.)  32.326  ft.,  29°  6'  35". 

(18.)  5.6569  miles  an  hour. 

(19.)  56.295  ft. 

(20.)    103.09  ft. 


(21.)  71°  33'  54". 

(22.)  858,160  miles. 

(23.)  238,850  miles. 

(24.)  2163.4  miles. 

(25.)  90,824,000  miles. 

(26.)  432.08  ft. 

(27.)  60.191  ft. 

(28.)  0.32149  mile. 

(29.)  193.77  ft. 

§  79  (page  83). 
(i.)  3416  ft. 

(2.)    3.7865  ft. 
(3.)    20.45  ft- 

(4.)  36.024^. 
(5.)  8.6058  sq.  ft. 
(6.)  181.23  in. 

(70  2-9943  ft. 
(8.)  5.1311  in. 
(9.)  25.92  ft. 
(io.)  92°  i'  24", 
(11.)  1.2491. 
(12.)  33°  12'  4". 
(13.)  11248  ft. 
(14.)  0.60965  miles. 
(15.)  1.3764. 
(16.)  1.9755- 
(17.)  19.882. 
(i 8.)  0.9397. 
(19.)  6.4984. 
(20.)  3.4641. 

(21.)  6.1981. 
(22.)  6.9978. 
(23.)  15.25. 

§  80  (page  84). 

(78.)   X  —  OO°,  I  20°,  2400,  270°. 

(79.)  ,r  =  o°,    20°,  45°,   90°,   100°, 

I35C,      140°,      1 80°,     220°, 

225°,    260°,    270°,    315°, 

340°. 


ANSWERS   TO  EXERCISES 


137 


(8o.)  ^-  =  0°,  30°,  90°,  150°,   1  80°, 

(24.)  55.74  ft. 

270°. 

(25.)  247.52  ft. 

(8  1.)  _r  =  o°,  45°,  120°,  240°,  225°, 

(26.)  556.34  ft. 

270°. 

(27.)  465.72  ft. 

(82.)  ;r  =  o0,  90°,  1  80°,  270°. 

(28.)  109.22  ft. 

(83.)  .r  =  0°,  90°,  210°,  330°. 

(29.)  2639.4  ft. 

(84.)  ^-  =  240°,  300°. 

(30.)  396-  54  ft. 

(85.)  x  =  2ioP,  330°- 

(31.)  287.75  ft. 

(86.)  x  =  o°,  90°. 

(32.)  2280.6  ft. 

.(87.)  ;r  =  o°,  1  80°. 

(33.)  64.62  ft. 

(88.)  *  =  o°,  1  80°. 

(34.)  127.98  ft. 

(89.)  ^  =  0°,  90°,  120°,  1  80°,  240°,      (35.)  45-l83  ft- 

270°. 

(36.)  4365-2  ft. 

(90.)  ^-  =  450,1350,2250,3150. 

(37.)  140.17  ft. 

(91.)  ^  =  30°,  150°,  210°,  330°. 

(38.)  610.45  ft- 

(39.)  1  56.66  ft. 

§  81  (page  88). 

(40.)  41°  48'  39"  and  125°  25 

(i.)  2145.1  ft. 

(41.)  51,288,000. 

(2.)  12.458  miles. 

(42.)  366680. 

(3.)  1.1033  miles. 

(43-)  U586. 

(4.)  1508.4  ft. 

(44.)  947460. 

(5-)  I7I9-3  yards. 

(45.)  0.89782. 

(6.)  1.2564  miles. 

(46.)  9929-3- 

(7.)   1346.3^. 

(47.)  7  5  1.  62  sq.ft. 

(8.)  387.1  yards. 

(48.)  3H5-9- 

(9.)  5.1083  miles. 

(49.)  855.1. 

(10.)  3791-8  ft. 

(50.)  876.34. 

(u.)  4.4152  ft- 
(12.)  28°  57'  20". 

§  88  (page  98). 

(13.)  115.27. 

(i.)   ^=54°  59'  47", 

(14.)  44.358  ft. 

^  =  45°  41  '28", 

(15.)  92.258  ft. 

s~*         /*     o       p>  '    rR'' 

(16.)  101°  32'  16". 

(2.)  C=7i°  36'  47". 

(17.)  0.83732  mile. 

^  =  95°  22'; 

(18.)  539.1  ft. 

c  =  7i°  32'  14", 

(19.)  1.239. 

(3.)  C=64°  1  4'  30", 

(20.)  152.31  and  238.3. 

C'—  115°  45'  3°  . 

(21.)  68.673  ft- 

.00    ^^'    rr" 

a  m  4^    22   55  > 

(22.)   32.071   ft. 

*'  =  i3i°37'  5". 

(23.)  13778ft. 

^=42°  19'  17". 

57' 


1 38 


ANSWERS   TO  EXERCfSES 


c  =  137°  40  43" 
(4.)  C- 65°  49' 54" 
a  =  63°  10'  6", 

£  =  38°  59'  12". 
(5.)  a  =  7S°  13' i", 
^=58°  25'  46", 

(6.)  a  =  76°  30' 37", 
£  =  65°  28'  58," 
<r  =  55°  47' 44". 


3' 
(8.) 

(9-) 


,  =  64°  36' 39", 
=  47°  57' 45" 
'-96°  1 3' 23", 
=  73°  1 7' 29", 
=  70°  8'  38". 
=  66°  58', 


(10.)  tf  =  6i°4'  55", 
b  =  40°  30'  22", 
<r=5o°  30'  32". 


§  99  (page  107). 


(2.)   a 
b 

(3-)  a 
(4.)  B 


131°  36' 36", 
116°  36'  38", 
29°  1 1' 42". 

107°  7' 45", 
48°  57'  29", 
62°  31  '40". 
62°  54'  43", 
114°  30'  26", 
=  56°  39'  10". 


a= 


(5.)  ^  =  130°  35'  56", 
^  =  30°  25'  34", 
C  =  3i°  26' 32", 

(6.)    ^=98°  21'  22", 

b-=.  109°  50'  8", 

*•=  115°  13'  4". 
(7.)  B  =  y.°  26'  9'', 

a=  84°  14'  32", 

^  =  51°  6'  12". 
(8.)  tf-8o°  5'  8", 

£  —  70°  10'  36", 

r  =  i45°5'2". 
(9.)  .4=70°  39'  4", 

#  =  48°  36'  2", 

C=ii9°  15'  2". 
(10.)  a  =  40°  o'  12", 

^  =  42°  15'  ii", 
C=i2i°  36'  19". 

§  100  (page  109). 

(i.)  80.895  sq-  in- 
(2.)  26.869  sq.  in. 
(3,)  158.41  sq.  in. 
(4.)  39990  sq.  miles. 

§  101  (page  112). 
(i.)  5C  =  48°  2' 43", 

^=52°  53' 9". 
(2.)  7  :  24  A.M. 

(3.)  4  P.M. 

§  102  (page  114). 
(I.)  3029^  miles. 
(2.)  2229.8  miles. 
(3.)  2748.5  miles. 
(4.)  7516.3  miles. 
(5.)  5108.9  miles. 


THE   END 


LOGARITHMIC 

AND 

TRIGONOMETRIC  TABLES 

FIVE-PLACE  AND  FOUR-PLACE 


PHILLIPS-LOOMIS  MATHEMATICAL  SERIES 


LOGARITHMIC 

AND 

TRIGONOMETRIC   TABLES 


FIVE-PLACE  AND  FOUR-PLACE 


BY 

ANDREW    W.    PHILLIPS,    PH.D. 

AM) 

WENDELL   M.  STRONG,  PH.D. 

YALE   UNIVERSITY 


NEW    YORK    AND    LONDON 

HARPER     &     BROTHERS     PUBLISHERS 
1899 


THE  PHILLIPS-LOOMIS  MATHEMATICAL  SERIES. 


ELEMENTS    OF    TRIGONOMETRY,    Plane    and    Spherical.      By 

ANDREW  W.  PHILLIPS,  Ph.D.,  and  WENDELL  M.  STRONG,  Ph.D.,  Yale 

University.    Crown  8vo,  Cloth. 
ELEMENTS  OF  GEOMETRY.     By  ANDREW  W.  PHILLIPS,  Ph.D., 

and  IRVING  FISHER,  Ph.D.,  Professors  in  Yale  University.     Crown 

8vo,  Half  Leather,  $1  75.    [By  mail,  $1  92.] 

ABRIDGED  GEOMETRY.  By  ANDREW  W.  PHILLIPS,  Ph.D.,  and 
IRVING  FISHER,  Ph.D.  Crown  8vo,  Half  Leather,  $1  25.  [By 
mail,  $1  40.] 

PLANE  GEOMETRY.  By  ANDREW  W.  PHILLIPS,  Ph.D.,  and  IRVING 
FISHER,  Ph.D.  Crown  8vo,  Cloth,  80  cents.  {By  mail,  90  cents.] 

LOGARITHMIC  AND  TRIGONOMETRIC  TABLES.  Five  Place 
and  Four- Place.  By  ANDREW  W.  PHILLIPS,  Ph.D.,  and  WENDELL 
M.  STRONG,  Ph.D.,  Yale  University.  Crown  8vo. 

LOGARITHMS  OF  NUMBERS.  Five-Figure  Table  to  Accompany 
the  "Elements  of  Geometry,"  by  ANDREW  W.  PHILLIPS,  Ph.D.,  and 
IRVING  FISHER,  Ph.D.,  Professors  in  Yale  University.  Crown  8vo, 
Cloth,  30  cents.  [By  mail,  35  cents.] 

NEW    YORK    AND    LONDON  : 
HARPER    &    BROTHERS,   PUBLISHERS. 


Copyright,  1898,  by  HARPER  &  BROTHERS. 


All  rights  reserved. 


CONTENTS 


TABLE  pAGE 

INTRODUCTION  TO  THE  TABLES v 

I.  FIVE-PLACE  LOGARITHMS  OF  NUMBERS i 

II.  FIVE -PLACE    LOGARITHMS    OF   THE   TRIGONOMETRIC 

FUNCTIONS  TO  EVERY  MINUTE 29 

III.  FIVE-PLACE  LOGARITHMS  OF  THE  SINE  AND  TANGENT 

OF  SMALL  ANGLES 121 

IV.  FOUR-PLACE  NAPERIAN  LOGARITHMS 131 

V.  FOUR-PLACE  LOGARITHMS  OF  NUMBERS 135 

VI.  FOUR -PLACE    LOGARITHMS    OF   THE  TRIGONOMETRIC 

FUNCTIONS  TO  EVERY  TEN  MINUTES 139 

VII.  FOUR -PLACE   NATURAL  TRIGONOMETRIC   FUNCTIONS 

TO  EVERY  TEN  MINUTES 149 

VIII.  SQUARES  AND  SQUARE  ROOTS  OF  NUMBERS     159 

IX.  THE  HYPERBOLIC  AND  EXPONENTIAL  FUNCTIONS  OF 

NUMBERS  FROM  o  TO  2.5  AT  INTERVALS  OF  .1  .    .  160 
X.  CONSTANTS— MEASURES   AND   WEIGHTS   AND  OTHER 

CONSTANTS    .        161 


INTRODUCTION  TO  THE  TABLES 

COMMON    LOGARITHMS. 

1.  The  common  logarithm  of  a  number  is  the  index  of 
the  power  to  which  10  must  be  raised  to  give  the  number. 

Thus,  log  IOG    =  2,          because  100    =  io2 

log      i     =o,  "  i    =10° 

log        .1  =  —  i,  .1  =  io-I 

log     3    =47712,       "  3    =io-47m 

In  general,  log  m  —  x  if  ;;/  =  io*. 

2.  To  multiply  two  numbers,  add  their  logarithms.     The 
result  is  the  logarithm  of  the  product. 

Proof. —  Ifaw  =  io*  so  that  log  m  =  x, 

and  n  =  io>  "  "  log  n  =y, 

then  mn  =  io*+*"  "  log  mn  =  x+y. 

Hence  log  mn  =  \ogrn  -f  log  n. 

3.  To  divide  one   number  by  another,  subtract  the  loga- 
rithm of  the  divisor  from  the  logarithm  of  the  dividend. 
The  result  is  the  logarithm  of  the  quotient. 

Proof.—  —  =       • 

Hence  log^~ =x~? 

4.  To  raise  a  number  to  a  power,  multiply  the  logarithm 
of  the  number  by  the  index  of  the  power.     The  result  is  the 
logarithm  of  the  power. 


vi  INTRODUCTION   TO    THE    TABLES. 

Proof.  —  ma     =  ( i  ox)a  =  i  o-ax  ; 

Hence  \ogma  =  ax  =  a  logm. 

5.  To  extract  a  root  of  a  number,  divide  the  logarithm  of 
the  number  by  the  index  of  the  root.  The  result  is  the  loga- 
ritJini  of  the  root. 


Proof.—  "Im  =-.  *  Ao*  =  10*. 


*v^>s   A.      /       ""    ~  7      » 

b         b 
6*.  Restatement  of  laws  : 

log  nin  =  log  in + log  n ; 

log—  =  logm  —  logn ; 
log  ma  =  a  log  m ; 


7.  Most  numbers  are  not  integral  powers  of   10;    hence 
most  logarithms  are  of  decimal  form. 

Thus,  log  2. 2  — .34242,    Iog4  =1.60206. 

S.   If  a  logarithm  is  negative,  it  is  expressed  for  conven- 
ience as  a  negative  integer  plus  a  positive  decimal. 

The  logarithm  of  a  number  less  than  I  is  negative. 

The   negative  integer  is  usually  expressed   in   the  form 
9—10,  8—  10,  etc. 

Thus,  Iog.2i544  =  —  i  -f  .33333,  written  9-33333  -  10  : 
Iog.o2i544  =  —  2 -h. 33333,  "  8.33333—10; 
log  .0021 544  =  —  3  +  .33333,  "  7-33333  ~  10. 

Remark. — In  some  books  the  negative  integer  is  written  i,  2,  etc., 
instead  of  9—  10,  8—  10,  etc. 

The  integral  part  of  a  logarithm  is  the  characteristic; 
the  decimal  part  is  the  mantissa. 

Thus,  log  2 1 5.44  =  2. 33333  ;  the  characteristic  is  -f- 2  ;  the  mantissa 


COMMON  LOGARITHMS.  vii 

is  +-33333:   log  .021544=8.33333— 10 ;   the  characteristic  is  8— 10 
=  —  2  ;  the  mantissa  is  -f  .33333. 

9.  It  is  evident  that  the  larger  a  number  the  larger  its  logarithm. 
Hence  the  logarithm  of  any  number 

between    i       and    10      is      o -f- a  mantissa, 
10        "     100       "       i+"         " 
.1      "        i        "-i+" 
.01    "          .1     "  —  2-f"         "        etc. 
We  have,  then,  the  following  rule  for  obtaining  the  characteristic : 

10.  Count  the  number  of  places  the  first  left-hand  digit  of 
the  number  is  removed  from  the  unit's  place. 

If  this  digit  is' to  the  left  of  the  unit's  place,  the  result  is  the 
required  characteristic. 

If  this  digit  is  to  the  rig/it  of  the  unit's  place,  the  result 
taken  with  a  minus  sign  is  the  required  characteristic. 

If  this  digit  is  in  the  unit' s  place,  t/ic  characteristic  is  zero. 
Thus  the  characteristic  of  the  logarithm  of  21550  is      4 

"       "  "  '  "        21.55         ••       i 

2.155       "       o 

' -2155     "—i 

"          "  "  "       "  »  "  .02155    "    -2 

11.  The  logarithms  of  numbers  which  differ  only  in  the 
position  of  the  decimal  point  have  the  same  mantissa. 

For  to  change  the  position  of  the  decimal  point  is  to  multiply  or 
divide  by  an  integral  power  of  10;  that  is,  an  integer  is  added  to  or 
subtracted  from  the  logarithm,  and  consequently  only  the  character- 
istic is  changed. 

Thus,  log  2 1 544  =3-33333 

log       2.1544     =0.33333 
log         .21544   =9.33333-10 
log         .021544  =  8.33333-10 

Therefore,  in  finding  the  mantissa  of  the  logarithm  of  a 
number  the  decimal  point  may  be  disregarded.  The  man- 
tissa is  found  from  the  tables  of  logarithms. 


viii  INTRODUCTION   TO    THE    TABLES. 

USE   OF  THE   TABLE   OF   LOGARITHMS   OF   NUMBERS. 

(TABLE  i.) 

12.  To  find  the  logarithm  of  a  number. 

Look  in  the  column  at  the  head  of  which  is  "  N  "  for  the 
first  three  figures  of  the  number,  and  in  the  line  with  "N"  for 
the  fourth  figure.  In  the  line  opposite  the  first  three  figures 
and  in  the  column  under  the  fourth  is  the  desired  mantissa. 

Only  the  last  three  figures  of  the  mantissa  are  found  thus;  the 
first  two  must  be  taken  from  the  first  column  ;  they  are  found  either 
in  the  same  line  or  in  the  first  line  above  which  gives  the  whole  man- 
tissa, except  when  a  *  occurs.  If  a  *  precedes  the  last  three  figures  of 
the  mantissa  the  first  two  are  found  in  the  following  line : 

The  characteristic  is  obtained  by  §  10. 

Example. — To  find  the  logarithm  of  105400. 

The  characteristic  =  5.  §  10 

The  mantissa         =   .02284  (opposite  103  and  under  4  in  the  tables) ; 

Hence  log  105400  =  5.02284. 

13.  If  there   are  five  or   more  figures  in  a  number  the 
figures  beyond  the  fourth  are  treated  as  a  decimal.     The 
corresponding  mantissa  is  between  two  successive  mantissas 
of  the  tables. 

Example. — To  find  the  logarithm  of  10543. 

The  characteristic  =  4.  §  10 

The  mantissa  is  not  in  the  tables,  but  is  between  the  mantissa  of 

1055  =  .02325 
and  the  mantissa  of  1054  =  .02284 

Their  difference  =        41 

Hence  an  increase  of  one  in  the  fourth  figure  of  the  number  pro- 
duces an  increase  of  41  in  the  mantissa.  Then  an  increase  of  .3  must 
produce  an  increase  of  41  X  .3  in  the  mantissa. 

41  X. 3  =  12.3  =  12  nearly. 

Hence  the  mantissa  of  10543=1.02284-}-  12  =  .02296. 

Therefore  log  10543=  4.02296. 


LOGARITHMS  OF  NUMBERS.  ix 

An  easy  method  of  multiplying  41  by  .3  is  to  use  the  table  of  pro- 
portional parts  at  the  bottom  of  the  page  in  the  tables. 
Under  41  and  opposite  3  is  12. 3 (=41  X-3). 

14.  Figures  beyond  the  fifth  are  usually  omitted  in  the 
use  of  a  five -place  table,  as  their  retention  does  not  add 
much  to  the  accuracy  of  the  result.  For  the  fifth  figure, 
however,  we  choose  the  one  which  gives  most  nearly  the 
true  value  of  the  number. 

Thus,  if  the  number  is  157.032,  we  use  157.03; 

"   157.036,    "     "     157.04; 

"    "         "        "   157.035.   "     "     157-04. 

13.    To  find  a  number  from  its  logarithm. 
The  process  is  the  reverse  of  finding  the  logarithm  from 
the  number;   it  is  illustrated  by  the  following  examples: 
Find  the  number  of  which  9.12872  — 10  is  the  logarithm. 
Since  the  characteristic  =  —  i,  the  decimal  point  will  be  before  the 
first  figure  of  the  number. 

.12872  is  opposite  134  and  under  5  in  the  tables. 
Hence  .12872  =  the  mantissa  of  1345, 

and  9.12872— 10  =  log. 1 345. 

Find  the  number  of  which  9.12895  —  10  is  the  logarithm. 
The  mantissa  .12895  is  not  i"  the  tables,  but  is 
between  .12905  =  mantissa  of  1346 

and  .12872=        "          "  1345. 

.00033  = tne  difference. 
.12895  —  mantissa  given, 
.12872  =  mantissa  of  1345,  the  smaller  number, 

23  =  the  difference. 

Change  §§  into  a  decimal.     The  first  figure  of  this  decimal  will  be 
the  figure  in  the  fifth  place  of  the  number. 

§f  =  .7  nearly. 
Hence  9.12895  — 10  — log. 13457. 


x  INTRODUCTION    TO    THE   TABLES. 

An  easy  method  of  changing  §§  into  a  decimal  is  to  use  the  table 
of  proportional  parts. 

Under  33  is  found  23.1  (=  23  nearly),  which  is  opposite  7. 

Hence  H  =  -7  nearly. 

The  process  we  have  employed  in  finding  the  logarithm 
of  a  number  of  more  than  four  figures,  or  the  number  corre- 
sponding to  a  mantissa  not  given  in  the  table,  is  called  in- 
terpolation. 

EXAMPLES   FOR  THE   USE   OF   LOGARITHMS. 

16.  Multiply  5789.2  by  .018315. 

tog  5789.2  =  3.76262 

log  .018315  =8.26281  —  10 

2.02543  =  log  106.03 
Multiply  9.8764  by  .10013. 

log  9.8764  =  0.99460 
log.  10013  =  9.00056—  10 

9.99516  —  10  =  log  .98892 
Find  the  value  of  3.1416  X  7638.6  x  .017829. 
log  3.1416=0.49715 
log  7638.6  =  3.88302 
log  .017829  =  8.251 13—  10 

2.631 30  =  log  427. 86 
Divide  81.321  by  3.1416. 

Iog8i.3i2  =  1.91021 
log  3. 141 6  =  0.497 1 5 

1.41306  =log  25.886 
Find  the  value  of  (2.1345)'. 

log  2. 1 345  =0.32930 

5 

1.64650  =  log  44.310 
Find  the  value  ofy/.oio2i. 

log  .01021  =  8.00903  —  10 
=  28.00903  —  30 

28.00003  —  30 

^ — ^-=9.33634 -10  =  log. 21694 


LOGARITHMS  OF   TRIGONOMETRIC  FUNCTIONS,     xi 

17.  The  logarithm  of  —  is  called  the  cologarithm  of  m, 

and  is  obtained  by  subtracting  logm  from  zero. 

Thus,  if  log m  =  9.76423—  10,  cologw  =0.23577. 

It  is  frequently  shorter  to  add  cologm  than  to  subtract 
logm  when  we  wish  to  divide  by  a  number  m. 

The  following  example  illustrates  this : 

Find  the  value  of  57fX42'24. 
644.32 

log  57. 98=  1.76328 
log  42. 24=  1.62572 
colog  644.32  =  7.19090—  10 

0.57990  =  log  3.801 

USE  OF  THE  TABLE   OF  LOGARITHMS   OF  TRIGONOMETRIC 

FUNCTIONS.    (TABLE  n.) 

18.  For  an  angle  less  than  45°,  the  degrees  are  at  the 
head  of  the  page,  the  minutes  in  the  column  at  the  left,  and 
"L.  Sin.,"  "L.  Tang.,"  etc.,  at  the  head  of  the  correspond- 
ing columns.     For  angles  between  45°  and  90°,  the  degrees 
are  at  the  foot  of  the  page,  the  minutes  in  the  column  at 
the  right,  and  "  L.  Sin.,"  "  L.  Tang.,"  etc.,  at  the  foot  of  the 
corresponding  columns. 

The  characteristic  is  printed  10  too  large  where  it  would 
otherwise  be  negative.  Hence,  in  using  this  table,  — 10  is 
to  be  supplied,  except  for  the  cotangent  of  angles  less  than 
45°  and  the  tangent  of  angles  from  45°  to  90°. 

EXAMPLES. 

log  sin  15°  25' =  9.42461  — 10. 
log  tan  28°  1 7' =  9.73084— 10. 
log  cos  62°  14'  =  9.66827  —  10. 
log  cot  25°  34' =  0.3  2020. 


xii  INTRODUCTION    TO    THE    TABLES. 

19.  If  the  given  angle  contains  seconds,  we  may  reduce 
the  seconds  to  a  decimal  of  a  minute  and  proceed  as  in 
finding  the  logarithms  of  numbers.  It  must  be  remem- 
bered, however,  that  log  cos  and  log  cot  decrease  as  the 
angle  increases. 

In  practice  we  remember  that  6"  is  one-tenth  of  a  minute,  and  di- 
vide the  number  of  seconds  by  6",  then  use  the  table  of  proportional 
parts  at  the  bottom  of  the  page. 

EXAMPLES. 
Find  log  sin  28°  14'  36"  (=log  sin  28°  14.6'). 

log  sin  28°  1  5'  —  log  sin  28°  14'  =  23  (found  in  column  "d.") 
log  sin  28°  14'  =  9.67492  —  10 
23  X.  6  =  13.8=  14  nearly 

log  sin  28°  14'  36"  =  9.67506—  10 

Find  log  cos  39°  17'  22"  (=log  cos  39°  17.3!'). 
log  cos  39°  1  7'  =  9.8887  5—  10 


log  cos  39°  17'  22"  =  9.88871  —  10 

Find  log  tan  51°  27'  44"  (=log  tan  51°  27.7^'). 
log  tan  51°  27'  =  .09862 


log  tan  51°  27'  44"  =  .0988  1 

Find  log  cot  67°  i8'46". 

log  cot  67°  1  8'  =9.62150  —  10 
6X.=          28 


Hence  log  cot  67°  18'  46"  =  9.62122  —  10 

20.  The  'process  of  finding  an  angle,  if  its  logarithmic 
sine  or  tangent,  etc.,  is  given,  is  the  reverse  of  the  pre- 
ceding. 


EXPLANATION  OF   THE    TABLES.  xiii 

EXAMPLES. 
Given  log  sin  x  =  9.67433  —  10 ;  find  x, 

log  sin  28°  ii'  =  9.6742 1  —  10 
log  sin ,r  — log  sin  28°  ii' =          12 
and        log  sin  28°  12'  — log  sin  28°  H' =          24 
Hence  .r  =  28°  1 1'  30"  (££  of  if  being  30"). 

Find  the  angle  whose  log  005  =  9.88231  —  10. 
log  cos  40°  1 8' =  9.88234—  10. 

60"  x  ft=  16". 
Hence  log  cos  40°  18'  1 6"  =  9.88231  — 10. 

Find  the  angle  whose  log  tan  =0.17844. 
log  tan  56°  27  =0.17839. 

6o"x&=u". 
Hence          '  log  tan  56°  27' ii"  =  0.17844. 

Find  the  angle  whose  log  cot  =  9.87432 — 10. 
log  cot  53°  10'  =  9.87448  —  10. 

6o"xi£=37"- 
Hence  log  cot  53°  10'  37"  =  9.87432—  10. 

EXPLANATION   OF  THE   TABLES. 

21.  A  dash  above  the  terminal  5  of  a  mantissa,  as  5,  de- 
notes that  the  true  value  is  less  than  5. 

Thus,  log  389  =  2.5899496  to  seven  places,  but  to  five  places 
log  389  =  2.58995. 

Tables  I  and  II  have  already  been  explained. 

TABLE  III. 

22.  The  logarithmic  sine  and  tangent  cannot  be  obtained 
very  accurately  from  Table  II  if  the  angle  contains  seconds 
and  is  less  than  2°. 

Table  III  is  to  be  used  when  greater  accuracy  in  the  sine 
or  tangent  of  a  small  angle  is  desired  than  can  be  obtained 


xivr  INTRODUCTION    TO    THE    TABLES. 

by  the  use  of  Table  II.  It  is  to  be  noted  that  the  first  page 
of  Table  III  gives  the  sine  and  tangent  to  every  second  for 
angles  less  than  8'. 

TABLE    IV. 

23.  Naperian  or  "natural"  logarithms  are  logarithms  to 
the  base  e  (  =  2.71828  +  ).      The  whole  logarithm  is  given, 
since  the  integral  part  cannot  be  supplied  by  inspection,  as 
with  common  logarithms. 

TABLES   V   AND   VI. 

24.  Four-place  logarithms  and  logarithmic  functions  are 
used  instead  of  five-place  if  the  results  are  sufficiently  ac- 
curate for  the  purpose  in  view. 

In  Table  VI  both  the  degrees  and  minutes  are  in  the  col- 
umns at  the  sides  of  the  page,  otherwise  this  table  does  not 
differ  in  form  from  Table  II. 

TABLE   VII. 

23.  This  table  is  identical  with  Table  VI  in  form,  but 
gives  the  trigonometric  functions  themselves,  instead  of 
their  logarithms. 

TABLES  VIII,   IX,   X. 

26.  These  tables  require  no  explanation. 


TABLE  I 

FIVE-PLACE    LOGARITHMS 
OF    NUMBERS 


100-130 


N 

O 

1 

2 

3 

4 

5 

O 

7 

8 

9 

100 

oo  ooo 

o43 

o87 

i3o 

i73 

2I7 

260 

3o3 

346 

389 

IOI 

432 

475 

5i8 

56i 

6o4 

647 

689 

732 

775 

817 

102 

860 

9o3 

945 

988 

*o3o 

*072 

*ii5 

*i57 

io3 

01 

284 

326 

368 

4io 

452 

494 

536 

578 

620 

662 

104 

7o3 

745 

787 

828 

87o 

912 

953 

995- 

*o36 

*o78 

io5 

02 

119 

1  60 

202 

243 

284 

325 

366 

407. 

449 

490 

106 

53i 

572 

612 

653 

694 

735 

776 

816 

857 

898 

107 

938 

979 

'9 

*o6o 

*IOO 

*i4i 

*i8i 

*222 

*262 

*302 

108 

o3 

342 

383 

423 

463 

5o3 

543 

583 

623 

663 

7o3 

109 

743 

782 

822 

862 

902 

94  1 

98i 

*O2I 

*o6o 

*IOO 

110 

o4  1  3g 

i79 

218 

258 

297 

336 

376 

415 

454 

493 

1  1  1 

532 

57i 

610 

650 

689 

727 

766 

8o5 

844 

883 

I  12 

922 

961 

999 

*o38, 

*°77 

*n5 

*i54 

*I92 

*23l 

*269 

n3 

o5 

3o8 

346 

385 

423 

46  1 

500 

538 

576 

6i4 

652 

ii 

4 

690 

729 

767 

805 

843 

881 

9i8 

956 

994 

*032 

ii 

5 

06  070 

1  08 

i45 

i83 

221 

258 

296 

333 

37i 

4o8 

116 

446 

483 

521 

558 

595 

633 

67o 

7°7 

744 

78i 

117 

819 

856 

893 

93o 

967 

*oo4 

*o4i 

*o78 

*ii5 

*i5i 

1x8 

07 

188 

225 

262 

298 

335 

372 

4o8 

445 

482 

5i8 

119 

555 

59i 

628 

664 

7oo 

737 

773 

809 

846 

882 

120 

918 

954 

99° 

*027 

*o63 

*°99 

*i35 

*i7i 

*2O7 

*a43 

121 

08 

279 

3i4 

35o 

386 

422 

458 

493 

529 

565 

600 

122 

636 

672 

707 

743 

778 

8i4 

849 

884 

92O 

^955 

123 

991 

*O26 

*o6i 

"096 

*l32 

*i67. 

*2O2 

*237 

*272 

124 

09 

342 

377 

4l2 

447 

482 

617 

552 

587 

621 

656 

125 

691 

726 

760 

796 

83o 

864 

899 

934 

968 

*oo3 

126 

10 

o37 

O72 

1  06 

i4o 

175 

209 

243 

278 

3l2 

346 

I27 

38o 

4i5 

449 

483 

5i7 

55i 

585 

619 

653 

687 

128 

•721 

755 

789 

823 

857 

890 

924 

958 

992 

*025 

I29 

ii 

o5g 

093 

126 

1  60 

i93 

227 

261 

294 

327 

36i 

130 

394 

428 

46  1 

494 

528 

56i 

594 

628 

661 

694 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

PP 

44 

43    42 

41    40    39 

38    37    36 

i 

4.4 

4.3   4.2 

i 

4.i 

4.o 

3.9 

i 

3.8 

3.7 

3.6 

2 

8.8 

8.6   8.4 

2 

8.2 

8.0 

7-8 

2 

7.6 

7-4 

7-2 

3 

13.2 

12.9   12.6 

3 

12.3 

I2.O 

n.7 

3 

n.4 

ii.  i 

10.8 

4 

17.6 

1-7.2   16.8 

4 

16.4 

16.0 

i5.6 

4 

15.2 

1  4.8 

i4.4 

5 

22.  0 

21.5   21.  0 

5 

20.  5 

20.  o 

i9.5 

5 

19.0 

i8.5 

18.0 

6 

26.4 

25.8   25.2 

6 

24.6 

24.0 

23.4 

6 

22.8 

22.2 

21.6 

7 

3o.8 

3o.i   29.4 

7 

28.7 

28.0 

27.3 

7 

26.6 

25.9 

25.2 

8 

35.2 

34.4   33.6 

8 

32.8 

32.0 

3l.2 

8 

3o.4 

29.6 

28.8 

9 

39.6 

9 

36.  9 

36.o 

35.i 

9 

34.2 

32.4 

13O— 16O 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

130 

1  1 

394 

428 

46i 

494 

528 

56i 

594 

628 

661 

694 

i3 

i 

727 

76o 

793 

826 

860 

893 

926 

959 

992 

*O24 

I  32 

12  057 

090 

123 

i56 

189 

222 

254 

287 

320 

352 

i33 

385 

4i8 

45o 

483 

5i6 

548 

58  1 

6i3 

646 

678 

1  34 

710 

743 

775 

808 

840 

872 

9°5 

937 

969 

*OOI 

i35 

i3 

o33 

066 

098 

i3o 

162 

i94 

226 

258 

290 

322 

1  36 

354 

386 

4i8 

45o 

48  1 

5i3 

545 

577 

6o9 

64o 

i37 

672 

704 

735 

767 

799 

83o 

862 

893 

925 

956 

1  38 

988 

*c 

'9 

*o5i 

*082 

*n4 

*i4g 

*i76 

*208 

*270 

139 

i4 

3oi 

333 

364 

395 

426 

457 

489 

52O 

55i 

582 

140 

6i3 

644 

675 

7o6 

737 

768 

799 

829 

860 

89i 

U 

i 

922 

953 

983 

*oi4 

+045 

*o76 

*io6 

*i37 

*i68 

*i98 

142 

i5 

229 

259 

290 

320 

35i 

38i 

412 

442 

473 

5o3 

M 

3 

534 

564 

594 

625 

655 

685 

7i5 

746 

776 

806 

i44 

836 

866 

897 

927 

957 

987 

*OI7 

*o47 

*077 

*IO7 

i45 

16 

i37 

167 

i97 

227 

256 

286 

3i6 

346 

376 

4o6 

i46 

435 

465 

495 

524 

554 

584 

6i3 

643 

673 

702 

i47 

732 

761 

7< 

pi 

820 

850 

879 

909 

938 

967 

997 

i48 

i7 

026 

o56 

085 

n4 

i43 

i73 

202 

23l 

260 

289 

149 

3i9 

348 

377 

4o6 

435 

464 

493 

522 

•55i 

58o 

150 

609 

638 

667 

696 

725 

754 

782 

811 

84o 

869 

i5 

i 

898 

926 

955 

984 

*oi3 

*o4i 

"070 

*o99 

*I27 

*i56 

i52 

18 

1  84 

2l3 

241 

270 

298 

327 

355 

384 

4l2 

44  1 

i53 

469 

498 

526 

554 

583 

611 

639 

6t>7 

696 

724 

1  54 

752 

780 

808 

837 

865 

893 

921 

949 

977 

*oo5 

i55 

19  o33 

06  1 

o89 

117 

i45 

i73 

201 

229 

257 

285 

i56 

312 

34o 

368 

396 

424 

45i 

479 

5o7 

535 

562 

i57 

590 

618 

645 

673 

700 

728 

756 

783 

811 

838 

1  58 

866 

893 

921 

948 

9-76 

*oo3 

*o3o 

*o58 

*o85 

*II2 

i59 

20 

i4o 

167 

i94 

222 

249 

276 

3o3 

33o 

358 

385 

160 

412 

439 

466 

493 

520 

548 

575 

602 

629 

656 

N 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

PP 

35 

34 

33 

32    31    30      29    28    27 

i 

3.5 

3.4 

3.3 

i 

3.c 

3.1 

3.o  i   2.9 

2.8 

2.7 

2 

7.0 

6.8 

6.6 

2 

6.4 

6^2 

6.0  2   5.8 

5.6 

5.4 

3 

io.5 

IO.2 

9-9 

3 

9.6 

9-3 

9.0  3   8.7 

8.4 

8.1 

4 

i4.o 

i3.6 

13.2 

4 

12.8 

12.4 

12.  o  4  ii.  6 

I  1.  2 

10.8 

5 

i7.5 

I7.0 

i6.5 

5 

16.0 

i5.5 

i5.o  $  i4.5 

i4.o 

i3.5 

6 

21.  0 

20.4 

19.8 

6 

19.2 

18.6 

18.0  6  17.4 

16.8 

16.2 

7 

24.5 

.23.8 

23.1 

7 

22.4 

21,7 

21.  o  7  20.  3 

19.6 

18.9 

8 

28.0 

27.2 

26.4 

8 

25.6 

24.8 

24.0   8   23.2 

22.4 

21.6 

° 

3i.5 

29.7 

9 

s8.8 

57.9 

27.0  9  26.1 

25.2 

24.3 

16O-190 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

160 

20  4l2 

439 

466 

493 

520 

548 

575 

602 

629 

656 

161 

683 

710 

737 

763 

79° 

817 

844 

871 

898 

925 

162 

952 

978 

*oo5 

*032 

*o59 

*o85 

*II2 

*:39 

*i65 

*I92 

i63 

21 

219 

245 

272 

299 

325 

352 

378 

405 

43i 

458 

1  64 

484 

5u 

537 

564 

59o 

617 

643 

669 

696 

722 

i65 

748 

775 

801 

827 

854 

880 

906 

932 

958 

985 

166 

22 

on 

037 

o63 

089 

u5 

i4i 

167 

i  $4 

220 

246 

167 

272 

298 

324 

350 

376 

4oi 

427 

453 

479 

5o5 

168 

53i 

557 

583 

608 

634 

660 

686 

712 

737 

763 

169 

7% 

8i4 

84o 

866 

891 

917 

943 

968 

994 

*oi9 

170 

23  045 

070 

096 

121 

i47 

172 

198 

223 

249 

274 

171 

3oo 

325 

35o 

376 

4oi 

426 

452 

477 

502 

5a8 

172 

553 

578 

6o3 

629 

654 

679 

704 

729 

754 

779 

i73 

805 

83o 

855 

880 

9°5 

93o 

955 

980 

*oo5 

*o3o 

1  74 

24 

055 

080 

105 

i3o 

155 

180 

204 

229 

254 

379 

i75 

3o4 

329 

353 

378 

4o3 

428 

452 

477 

502 

537 

176 

55i 

576 

601 

625 

650 

674 

699 

724, 

,748 

773 

177 

797 

822 

846 

871 

895 

920 

944 

969 

993" 

*oi8 

178 

25 

042 

066 

091 

n5 

i39 

1  64 

188 

212 

237 

261 

179 

285 

3io 

334 

358 

382 

4o6 

43i 

455 

479 

5o3 

180 

527 

55i 

575 

600 

624 

648 

672 

696 

720 

744 

181 

768 

792 

816 

84o 

864 

888 

912 

935 

959 

983 

182 

26 

007 

o3i 

055 

079 

1  02 

126 

i5o 

1  74 

198 

221 

i83 

245 

269 

293 

3i6 

34o 

364 

387 

4n 

435 

458 

1  84 

482 

5o5 

529 

553 

576 

600 

€23 

647 

670 

694 

i85 

717 

74i 

764 

788 

811 

834 

858 

881 

9°5 

928 

186 

95i 

975 

998 

*02I 

*o45 

*o68 

*09i 

*u4 

*i38 

*i6i 

187 

27 

1  84 

207 

23l 

254 

277 

3oo 

323 

346 

37o 

393 

1  88 

4i6 

439 

462 

485 

5o8 

53i 

554 

577 

600 

623 

189 

646 

669 

692 

7i5 

738 

761 

784 

807 

83o 

85a 

190 

875 

898 

921 

944 

967 

989 

*OI2 

*o35 

*o58 

*o8i 

N 

O 

1 

2 

3 

4 

5 

6 

7 

8    9 

PP  27 

26 

25 

24    23    22 

21    20    19 

i 

2-7 

2.6 

2.5 

: 

2.4 

2.3 

2.2 

i 

2.1 

2.O 

1.9 

2 

5.4 

5.2 

5.o 

2 

4.8 

4.6 

4.4 

2 

4.2 

4.o 

3.8 

3 

8.1 

7.8 

7-5 

3 

7.2 

6.9 

6.6 

3 

6.3 

6.0 

5-7 

4 

10.8 

10.4 

IO.O 

4 

9.6 

9.2 

8.8 

4 

8.4 

8.0 

7.6 

5 

i3.5 

i3.o 

12.5 

5 

12.0 

ii.5 

n.o 

5 

io.5 

IO.O 

9-5 

6 

16.2 

i5.6 

i5.o 

6 

i44 

i3.8 

13.2 

6 

12.6 

I2*.0 

n.4 

7 

18.9 

18.2 

i7.5 

7 

16.8 

16.1 

i5.4 

7 

i4.7 

i4.o 

i3.3 

8 

21.6 

20.8 

20.  o 

8 

19.2 

18.4 

17.6 

8 

16.8 

16.0 

15.2 

9 

24.3 

23.4 

22.5 

9 

21.6 

20.7 

19.8 

9 

18.9 

18.0   17.1 

190-230 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

190 

27876 

898 

921 

944 

967 

989 

*OI2 

*o35 

*o58 

*o8i 

191 

28  io3 

126 

149 

171 

194 

2I7 

240 

262 

285 

3o7 

192 

33o 

353 

375 

398 

421 

443 

466 

488 

5n 

533 

i93 

556 

578 

601 

623 

646 

668 

691 

7i3 

735 

758 

194 

780 

8o3 

825 

847 

87o 

892 

914 

937 

959 

98i 

i95 

29  oo3 

026 

o48 

O7O 

092 

115 

i37 

i59 

181 

203 

196 

226 

248 

270 

292 

3i4 

336 

358 

38o 

4o3 

425 

197 

447 

469 

491 

5i3 

53s 

557 

579 

601 

623 

645 

198 

667 

688 

710 

732 

754 

776 

798 

820 

842 

863 

199 

885 

9°7 

929 

95i 

973 

994 

*oi6 

*o38 

*o6o 

*o8i 

200 

3o  io3 

125 

i46 

1  68 

190 

211 

233 

255 

276 

298 

201 

32O 

34  1 

363 

384 

4o6 

428 

449 

47i 

492 

5i4 

202 

535 

557 

578 

600 

621 

643 

664 

685 

707 

728 

203 

750 

771 

792 

8i4 

835 

856 

878 

899 

920 

942 

2O4 

963 

984 

*oo6 

*027 

*o48 

*o69 

*O9i 

*II2 

*i33 

*i54 

205 

3i  i75 

197 

218 

239 

260 

281 

302 

323 

345 

366 

206 

387 

4o8 

429 

450 

47i 

492 

5i3 

534 

555 

576 

207 

597 

618 

639 

660 

681 

702 

723 

744 

765 

785 

208 

806 

827 

848 

869 

890 

911 

93i 

962 

973 

994 

209 

32  015 

o35 

o56 

077 

098 

118 

i39 

160 

181 

2OI 

210 

222 

243 

263 

284 

305 

325 

346 

366 

387 

4o8 

211 

428 

449 

469 

49o 

5io 

53i 

552 

572 

593 

6i3 

212 

634 

654 

675 

695 

7i5 

736 

756 

777 

797 

818 

2l3 

838 

858 

879 

899 

919 

94o 

96o 

980 

*OOI 

*O2I 

2l4 

33o4i 

062 

082 

102 

122 

i43 

i63 

i83 

203 

224 

2l5 

244 

264 

284 

3o4 

325 

345 

36s 

385 

4o5 

425 

216 

445 

465 

486 

5o6 

52.6 

546 

566 

586 

606 

626 

217 

646 

666 

686 

706 

726 

746 

766 

786 

806 

826 

218 

846 

866 

885 

906 

925 

945 

965 

985 

*oo5 

*O25 

219 

34o44 

o64 

o84 

io4 

124 

i43 

i63 

i83 

203 

223 

220 

242 

262 

282 

3oi 

321 

34i 

36i 

38o 

4oo 

420 

221 

439 

459 

4?9 

498 

5i8 

537 

557 

577 

596 

616 

222 

635 

655 

674 

694 

7i3 

733 

753 

772 

792 

811 

223 

83o 

850 

869 

889 

908 

928 

947 

967 

986 

*oo5 

224 

35  025 

o44 

o64 

o83 

102 

122 

i4i 

1  60 

1  80 

I99 

225 

218 

238 

257 

276 

295 

315 

334 

353 

372 

392 

226 

4n 

43o 

449 

468 

488 

5o7 

526 

545 

564 

583 

227 

6o3 

622 

64i 

660 

679 

698 

717 

736 

755 

774 

228 

793 

8i3 

832 

85i 

87o 

889 

908 

927 

946 

965 

229 

984 

*oo3 

*O2I 

*o4o 

*o59 

*o78 

*097 

*n6 

*i35 

*i54 

230 

36  i73 

192 

21  I 

229 

248 

26-7 

286 

305 

324 

342 

X 

O 

1    2 

3 

4 

5 

6 

7 

8   9 

23O— 26O 


N 

0 

1 

2 

3 

4 

5 

0 

7 

8 

0 

230 

36  173 

192 

211 

229 

248 

267 

286 

3o5 

324 

342 

s3i 

36i 

38o 

399 

4i8 

436 

455 

474 

493 

5n 

53o 

232 

549 

568 

586 

605 

624 

642 

661 

680 

698 

717 

233 

736 

754 

773 

79I 

810 

829 

847 

866 

884 

9o3 

234 

922 

94o 

959 

977 

996 

*oi4 

*o33 

*o5i 

"070 

*o88 

235 

37  io7 

125 

1  44 

162 

181 

199 

218 

236 

254 

273 

236 

291 

3io 

328 

346 

365 

383 

4oi 

420 

438 

457 

237 

475 

493 

5n 

53o 

548 

566 

585 

6o3 

621 

639 

238 

658 

676 

694 

7I2 

73i 

749 

767 

785 

8o3 

822 

239 

84o 

858 

876 

894 

912 

93i 

949 

967 

985 

*oo3 

240 

38  021 

039 

057 

o75 

093 

112 

i3o 

1  48 

166 

1  84 

241 

202 

220 

238 

256 

274 

292 

3io 

328 

346 

364 

242 

382 

399 

4i7 

435 

453 

47i 

489 

507 

525 

543 

243 

56i 

578 

596 

6i4 

632 

650 

668 

686 

7o3 

72I 

244 

739 

757 

775 

792 

810 

828 

846 

863 

881 

899 

245 

9i7 

934 

952 

970 

987 

*oo5 

*023 

*o4i 

*o58 

*o76 

246 

39o94 

in 

I29 

i46 

1  64 

182 

199 

217 

235 

252 

247 

270 

287 

3o5 

322 

34o 

358 

375 

393 

4io 

428 

248 

445 

463 

48o 

498 

5i5 

533 

55o 

568 

585 

602 

249 

620 

637 

655 

672 

690 

707 

•724 

742 

759 

777 

250 

794 

811 

820. 

846 

863 

881 

898 

9i5 

933 

95o 

•Si 

967 

985 

*002 

*oi9 

*o37 

*o54 

*o 

7i 

*o88 

*io6 

*I23 

252 

4o  i^ 

0 

i57 

175 

I92 

209 

226 

243 

261 

278 

295 

253 

3l2 

329 

346 

364 

38i 

398 

4i5 

432 

449 

466 

254 

483 

5oo 

5i8 

535 

552 

569 

586 

6o3 

620 

637 

255 

654 

67i 

688 

705 

722 

739 

756 

773 

79° 

807 

256 

824 

84i 

858 

875 

892 

909 

926 

943 

960 

976 

257 

993 

*OIO 

*027 

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*078 

*°95 

*ii 

i 

*I28 

*i45 

258 

4i  162 

179 

i96 

212 

229 

246 

263 

280 

296 

3i3 

269 

33o 

347 

363 

38o 

397 

4i4 

43o 

447 

464 

48  1 

260 

497 

5i4 

53i 

547 

564 

58i 

597 

6i4 

63i 

647 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

PP 

19 

18    17         16 

15 

14 

i 

T^ 

1.8 

1.7      i   1.6 

7s" 

i.4 

2 

3'.8 

3.6 

3.4      2   3.2 

3'.o 

2.8 

3 

5-7 

5.4 

5.i      3   4.8 

4.5 

4.2 

4 

7.6 

7.2 

6.8     4   6.4 

6.0 

5.6 

5 

9.5 

9.0 

8.5      5   8.0 

7-5 

7.0 

6 

1  1.  4 

10.8 

10.2      6   9.6 

9.0 

8.4 

7 

i3.3 

12.6 

11.9      7   ii.  2 

io.5 

9.8 

8 

15.2 

i4.4 

i3.6      8   12.8 

12.0 

I  1.  2 

9 

17.1 

16.2   i5.3      9   r4.4 

i3.5 

12.6 

260-300 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

260 

4i  497 

5i4 

53i 

547 

564 

58i 

597 

6i4 

63i 

647 

261 

664 

681 

697 

714 

73i 

747 

764 

780 

797 

8i4 

262 

83o 

847 

863 

880 

896 

9i3 

929 

946 

963 

979 

263 

996 

*OI2 

*029 

*o45 

*o62 

*078 

*°95 

*in 

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^i44 

a64 

42  1  60 

177 

i93 

210 

226 

243 

259 

275 

292 

3o8 

265 

325 

34  1 

357 

374 

39o 

4o6 

423 

439 

455 

472 

266 

488 

5o4 

521 

537 

553 

-570 

586 

602 

6i9 

635 

267 

65i 

667 

684 

700 

716 

732 

?49 

765 

781 

797 

268 

8i3 

83o 

846 

862 

878 

894 

9u 

927 

943 

959 

269 

975 

991 

*oo8 

*024 

*o4o 

*o56 

*O72 

*o88 

*io4 

*I20 

270 

43  i36 

152 

169 

185 

201 

217 

233 

249 

265 

28l 

271 

297 

3i3 

329 

345 

36i 

377 

393 

4o9 

425 

44  1 

272 

457 

473 

489 

505 

521 

537 

553 

569 

584 

600 

273 

616 

632 

648 

664 

680 

696 

712 

727 

743 

759 

274 

775 

791 

807 

823 

838 

854 

870 

886 

9O2 

9i7 

275 

933 

949 

965 

981 

996 

*OI2 

*028 

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*o59 

*o75 

276 

44091 

107 

122 

i38 

1  54 

I70 

i85 

2OI 

217 

232 

277 

248 

264 

279 

295 

3n 

326 

342 

358 

373 

389 

278 

4o4 

420 

436 

45i 

467 

483 

498 

5i4 

529 

545 

279 

56o 

576 

592 

607 

623 

638 

654 

669 

685 

7oo 

280 

716 

73i 

747 

762 

778 

793 

8o9 

824 

84o 

855 

281 

871 

886 

902 

9i7 

932 

948 

963 

979 

994 

*OIO 

282 

46025 

o4o 

o56 

071 

086 

IO2 

117 

i33 

i48 

i63 

283 

179 

I94 

209 

225 

240 

255 

271 

286 

3oi 

3i7 

284 

332 

347 

362 

378 

393 

4o8 

423 

439 

454 

469 

285 

484 

500 

5'5 

53o 

545 

56i 

576 

59i 

606 

621 

286 

637 

652 

667 

682 

697 

712 

728 

743 

758 

773 

287 

788 

8o3 

818 

834 

849 

864 

879 

894 

9°9 

924 

288 

939 

954 

969 

984 

*ooo 

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*o3o 

*o45 

*o6o 

*075 

289 

46  090 

105 

I2O 

i3s 

150 

165 

1  80 

195 

2IO 

225 

290 

240 

255 

270 

285 

3oo 

315 

33o 

345 

359 

374 

291 

389 

4o4 

419 

434 

449 

464 

479 

494 

5o9 

523 

292 

538 

553 

568 

583 

598 

6i3 

627 

642 

657 

672 

293 

687 

702 

716 

73i 

746 

761 

776 

79° 

8o5 

820 

294 

83,5 

850 

864 

879 

894 

9°9 

923 

938 

953 

967 

296 

982 

997 

*OI2 

*O26 

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*O70 

*o85 

*IOO 

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296 

47  129 

1  44 

1*9 

I73 

188 

202 

217 

232 

246 

261 

297 

276 

290 

305 

3i9 

334 

349 

363 

378 

392 

4o7 

298 

422 

436 

45i 

465 

48o 

494 

5o9 

524 

538 

553 

299 

567 

582 

596 

611 

625 

64o 

654 

669 

683 

698 

300 

712 

727 

74i, 

^756 

770 

784 

799 

8i3 

828 

842 

N 

0 

1 

a 

3 

4 

5 

6 

7 

8 

9 

3OO— 33O 


N 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

800 

47712 

727 

74i 

756 

77° 

784 

799 

8i3 

828 

842 

3oi 

857 

871 

885 

9oo 

914 

929 

943 

958 

972 

986 

302 

48  ooi 

oi5 

029 

o44 

o58 

073 

087 

101 

116 

i3o 

3o3 

i44 

i59 

I73 

187 

202 

216 

230 

244 

259 

273 

3o4 

287 

302 

3i6 

33o 

344 

359 

373 

387 

4oi 

4i6 

3o5 

43o 

444 

458 

473 

487 

5oi 

5i5 

53o 

544 

558 

3o6 

572 

586 

601 

615 

629 

643 

657 

671 

686 

7oo 

3o7 

7i4 

728 

742 

756 

770 

785 

799 

8i3 

827 

84i 

3o8 

855 

869 

883 

897 

911 

926 

954 

968 

982 

309 

996 

*OIO 

*024 

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*052 

*o66 

*o8o 

*o94 

*io8 

*I22 

310 

49  1  36 

i5o 

1  64 

178 

192 

206 

220 

234 

248 

262 

3u 

276 

290 

3o4 

3i8 

332 

346 

36o 

374 

388 

4O2 

3l2 

4i5 

429 

443 

457 

471 

485 

499 

5i3 

527 

54i 

3i3 

554 

568 

582 

596 

610 

624 

638 

65i 

665 

679 

3i4 

693 

707 

721 

734 

748 

762 

776 

79° 

8o3 

817 

3i5 

83i 

845 

859 

872 

886 

900 

914 

927 

941 

^955 

3i6 

969 

982 

996 

*OIO 

*024 

*o37 

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,g? 

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3i7 

5o  106 

I2O 

i33 

i47 

161 

1  74 

188 

202 

2l5 

229 

3i8 

243 

256 

270 

284 

297 

3n 

325 

338 

352 

365 

3i9 

379 

393 

4o6 

420 

433 

447 

46i 

474 

488 

5oi 

320 

5^5 

520 

542 

556 

569 

583 

596 

610 

623 

637 

321 

65i 

664 

678 

691 

705 

718 

732 

745 

759 

772 

322 

786 

799 

8i3 

826 

84o 

853 

866 

880 

893 

9°7 

323 

920 

934 

947 

961 

974 

987 

*OOI 

*oi4 

*028 

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324 

5i  055 

068 

08  1 

095 

1  08 

121 

138 

i48 

162 

'75 

325 

1  88 

202 

2l5 

228 

242 

255 

268 

282 

295 

3o8 

326 

322 

335 

348 

362 

375 

388 

402 

4i5 

428 

44  1 

327 

455 

468 

48  1 

495 

5o8 

521 

534 

548 

56i 

574 

328 

587 

601 

6i4 

627 

64o 

654 

667 

680 

693 

7o6 

329 

720 

733 

746 

759 

772 

786 

799. 

812 

825 

838 

330 

85i 

865 

878 

891 

904 

917 

93o 

943 

957 

97° 

N 

0 

1 

2 

3 

4 

5 

0 

7 

8    t> 

PP     15    14    13            12 

11 

! 

i.5 

i.4 

i.3 

I     1.2 

i.i 

2 

S'.Q 

2!8 

•*.6 

2     2.4 

2.2 

3 

4.5 

4.2 

3.9 

3   3.6 

3.3 

4 

6.0 

5.6 

5.2 

4   4.8 

4.4 

5 

7-5 

7.0 

6.5 

5   6.0 

5.5 

6 

9.0 

8.4 

7.8 

6   7.2 

6.6 

7 

io.5 

9-8 

9.1 

7   8.4 

7-7 

8 

12.  0 

II.  2 

10.4 

8   9.6 

8.8 

9 

i3.5 

12.6 

11.7 

9   10.8 

9-9 

330-370 


Jg 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

330 

5i  85i 

865 

878 

891 

904 

917 

93o 

943 

957 

97° 

33i 

983 

996 

*oo9 

*022 

*o35 

*o48 

*o6i 

*°75 

*o88 

*IOI 

332 

52  u4 

127 

i4o 

i53 

166 

179 

192 

2O5 

218 

23l 

333 

244 

257 

270 

284 

297 

3io 

323 

336 

349 

362 

334 

375 

388 

4oi 

4i4 

427 

44o 

453 

466 

479 

492 

335 

5o4 

5i7 

53o 

543 

556 

569 

582 

595 

608 

621 

336 

634 

64? 

660 

673 

686 

699 

711 

•724 

737 

75o 

337 

763 

776 

789 

802 

815 

827 

84o 

853 

866 

879 

338 

892 

9°5 

917 

930 

943 

956 

969 

982 

994 

*OO7 

339 

53  020 

o33 

o46 

o58 

071 

o84 

097 

I  IO 

122 

i35 

340 

i48 

161 

i73 

186 

199 

212 

224 

237 

250 

263 

34i 

275 

288 

3oi 

3i4 

326 

339 

352 

364 

377 

39o 

342 

4o3 

4i5 

428 

44  1 

453 

466 

479 

49i 

5o4 

5i7 

343 

529 

542 

555 

567 

58o 

593 

6o5 

618 

63i 

643 

344 

656 

668 

681 

694 

706 

719 

732 

744 

757 

769 

345 

782 

794 

807 

820 

832 

845 

857 

870 

882 

895 

346 

908 

920 

933 

945 

958 

97° 

983 

995 

*oo8 

*O2O 

347 

54o33 

o45 

o58 

070 

o83 

o95 

1  08 

120 

i33 

i45 

348 

1  58 

170 

i83 

i95 

208 

220 

233 

245 

258 

270 

349 

283 

295 

3o7 

320 

332 

345 

357 

370 

382 

394 

350 

407 

419 

432 

444 

456 

469 

48  1 

494 

5o6 

5i8 

35i 

53i 

543 

555 

568 

58o 

593 

605 

617 

63o 

642 

352 

654 

667 

679 

691 

7o4 

716 

728 

74  1 

753 

765 

353 

111 

79° 

802 

8i4 

827 

839 

85i 

864 

876 

888 

354 

900 

9i3 

925 

937 

949 

962 

974 

986 

998 

*OII 

355 

55  023 

o35 

047 

060 

O72 

o84 

096 

108 

121 

i33 

356 

i45 

i57 

169 

182 

194 

206 

218 

230 

242 

255 

357 

267 

279 

291 

3o3 

3i5 

328 

34o 

352 

364 

376 

358 

388 

4oo 

4i3 

425 

437 

449 

46i 

473 

485 

497 

359 

5og 

522 

534 

546 

558 

57o 

582 

594 

606 

618 

360 

63o 

642 

654 

666 

678 

691 

7o3 

7i5 

727 

739 

36i 

75i 

763 

775 

787 

799 

8n 

823 

835 

847 

859 

362 

871 

883 

895 

907 

919 

93i 

943 

955 

967 

979 

363 

991 

*oo3 

*oi5 

*027 

*o38 

*o5o 

*062 

*o74 

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364 

56  no 

122 

1  34 

i46 

i58 

170 

182 

194 

2O5 

217 

365 

229 

24  1 

253 

265 

277 

289 

3oi 

3l2 

324 

336 

366 

348 

36o 

372 

384 

396 

407 

419 

43i 

443 

455 

367 

467 

478 

490 

502 

5i4 

526 

538 

549 

56i 

573 

368 

585 

597 

608 

620 

632 

644 

656 

667 

679 

691 

369 

7o3 

7i4 

726 

738 

750 

761 

773 

785 

797 

808 

370 

820 

832 

844 

855 

867 

879 

891 

902 

9i4 

926 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

37O-4OO 


N 

0 

1 

2 

3 

4 

5 

0 

7 

8 

9 

370 

56  820 

832 

844 

855 

867 

879 

891 

902 

914 

926 

37i 

937 

949 

961 

972 

984 

996 

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372 

57o54 

066 

078 

089 

101 

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124 

1  36 

1  48 

i59 

373 

171 

i83 

1  94 

206 

217 

229 

241 

252 

264 

276 

374 

287 

299 

3io 

322 

334 

345 

357 

368 

38o 

392 

375 

4o3 

426  438 

449 

46  1 

473 

484 

496 

5o7 

376 

5i9 

53o 

542 

553 

565 

576 

588 

600 

611 

623 

377 

634 

646 

657  669 

680 

692 

7o3 

715 

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738 

378 
379 

749 
864 

76i 

875 

887 

784 
898 

795 
910 

807 
921 

818 
933 

83o 
944 

84i 
955 

852 
967 

380 

978 

99° 

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38i 

58  o92 

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n5 

127 

1  38 

149 

161 

172 

1  84 

J95 

382 

206 

218 

229 

240 

252 

263 

274 

286 

297 

309 

383 

320 

33i 

343 

354 

365 

377 

388 

399 

4io 

422 

384 

433 

444 

456 

467 

478 

490 

5oi 

5l2 

524 

535 

385 

546 

557 

569 

58o 

59i 

602 

6i4 

625 

636 

647 

386 

659 

67o 

681 

692 

704 

715 

726 

737 

749 

760 

387 

77i 

782 

794 

805 

816 

827 

838 

850 

861 

872 

388 

883 

894 

906 

917 

928 

939 

95o 

961 

973 

984 

389 

995 

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*062 

*°73 

*o84 

*095 

390 

59  1  06 

118 

129 

i4o 

i5i 

162 

i73 

1  84 

i95 

207 

39i 

218 

229 

240 

25l 

262 

273 

284 

295 

3o6 

3i8 

392 

329 

34o 

35i 

362 

373 

384 

395 

4o6 

4i7 

428 

393 

439 

45o 

46i 

472 

483 

494 

5o6 

5i7 

528 

539 

3g4 

550 

56r 

572 

583 

594 

605 

616 

627 

638 

64g 

395 

660 

671 

682 

693 

764 

7i| 

> 

•726 

737 

748 

759 

396 

77° 

780 

791 

802, 

8i3 

824 

835 

846 

857 

868 

397 

879 

890 

901 

912 

923 

934 

945 

956 

966 

977 

398 

988 

999 

*OIO 

*O2I 

*032 

*o43 

*o54 

"065 

*o76 

*o86 

399 

60  o97 

1  08 

119 

i3o 

i4i 

l52 

i63 

I73 

1  84 

i95 

400 

206 

217 

228 

239 

249 

260 

271 

282 

293 

3o4 

N 

0 

1 

2 

3 

4 

5 

0 

7 

8 

9 

PP     12     11 

10     9 

I 

1.2       1,1             I 

i.o     0.9 

2 

2.4      2.2             2 

2,0       1.8 

3 

3.6     3.3         3 

3.o     2.7 

4 

4.8    4.4        4 

4.o     3.6 

5 

6.0    5.5        5 

5.o     4.5 

6 

7.2     6.6         6 

6.0     5.4 

7 

8.4     7-7         7 

7.0     6.3 

8 

9.6    8.8         8 

8.0     7.2 

9 

10.8     9.9         9 

9.0     8.1 

10 


400-440 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

400 

60  206 

217 

228 

239 

249 

260 

271 

282 

293 

3o4 

4oi 

3i4 

325 

336 

347 

358 

369 

379 

390 

4oi 

412 

402 

423 

433 

444 

455 

466 

477 

487 

498 

509 

520 

4o3 

53i 

54i 

552 

563 

574 

584 

595 

606 

617 

627 

4o4 

638 

649 

660 

670 

681 

692 

7o3 

7i3 

724 

735 

4o5 

746 

756 

767 

778 

788 

799 

810 

821 

83i 

842 

4o6 

853 

863 

874 

885 

895 

906 

917 

927 

938 

949 

4oy 

959 

970 

981 

991 

*OO2 

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4o8 

6  1  066 

077 

087 

098 

I09 

119 

i3o 

i4o 

i5i 

162 

409 

172 

i83 

194 

204 

215 

225 

236 

247 

257 

268 

410 

278 

289 

3oo 

3io 

321 

33i 

342 

352 

363 

374 

4n 

384 

395 

4o5 

4i6 

426 

437 

448 

458 

469 

479 

412 

4go 

5oo 

5n 

621 

532 

542 

553 

563 

574 

584 

4i3 

5,5 

606 

616 

627 

637 

648 

658 

669 

679 

690 

4i4 

7OO 

711 

721 

73i 

742 

752 

763 

773 

784 

794 

4i5 

805 

8i5 

826 

836 

847 

857 

868 

878 

888 

899 

4i6 

909 

920 

93o 

94i 

95i 

962 

972 

982 

993 

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4i7 

62  oi4 

024 

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045 

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066 

076 

086 

097 

I07 

4i8 

118 

128 

i38 

149 

i59 

170 

1  80 

190 

2OI 

211 

419 

221 

232 

242 

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in 

118 

125 

132 

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i47 

1  54 

161 

168 

6o5 

176 

i83 

190 

i97 

204 

211 

219 

226 

233 

240 

606 

247 

254 

262 

269 

276 

283 

290 

297 

305 

3l2 

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326 

333 

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347 

355 

362 

369 

376 

383 

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597 

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0 

1 

2 

3 

4 

5 

6 

7 

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7              6 

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0.8 

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0.7 

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6 

4.8 

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7 

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4-9 

7 

4.2 

8 

6.4 

8 

5.6 

8 

4.8 

9 

7-2 

9 

6.3 

9 

5.4 

16 


010-650 


N 

0 

1 

2 

3 

4 

5 

6 

7 

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610 

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54o 

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554 

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569 

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583 

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618 

625 

633 

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647 

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661 

668 

612 

675 

682 

689 

696 

704 

711 

718 

725 

732 

739 

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746 

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760 

767 

774 

781 

789 

796 

8o3 

810 

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817 

824 

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838 

845 

852 

859 

866 

873 

880 

6i5 

888 

895 

902 

909 

916 

923 

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937 

944 

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616 

958 

965 

972 

979 

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993 

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617 

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071 

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169 

176 

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190 

197 

204 

211 

218 

225 

232 

620 

239 

246 

253 

260 

267 

274 

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288 

295 

302 

621 

309 

3i6 

323 

33o 

337 

344 

35i 

358 

365 

372 

622 

379 

386 

393 

4oo 

407 

4i4 

421 

428 

43s 

442 

623 

449 

456 

463 

470 

477 

484 

491 

498 

505 

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624 

5i8 

525 

532 

539 

546 

553 

56o 

567 

574 

58i 

626 

588 

595 

602 

609 

616 

623 

63o 

637 

644 

65o 

626 

657 

664 

671 

678 

685 

692 

699 

706 

7i3 

720 

627 

727 

734 

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748 

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761 

768 

775 

782 

789 

628 

796 

8o3 

810 

817 

824 

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837 

844 

85i 

858 

629 

865 

872 

879 

886 

893 

900 

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920 

927 

630 

934 

941 

948 

955 

962 

969 

975 

982 

989 

996 

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017 

024 

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065 

632 

072 

079 

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092 

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1  06 

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120 

127 

1  34 

633 

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1  54 

161 

168 

175 

182 

1  88 

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202 

634 

209 

216 

223 

229 

236 

243 

25o 

257 

264 

271 

635 

277 

284 

291 

298 

3o5 

312 

3i8 

325 

332 

339 

636 

346 

353 

359 

366 

373 

38o 

387 

393 

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407 

637 

4i4 

421 

428 

434 

44  1 

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462 

468 

475 

638 

482 

489 

496 

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509 

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523 

53o 

536 

543 

639 

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557 

564 

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577 

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598 

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611 

640 

618 

625 

632 

638 

645 

652 

659 

665 

672 

679 

64  1 

686 

693 

699 

706 

7i3 

720 

726 

733 

740 

747 

642 

754 

760 

767 

774 

781 

787 

794 

801 

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8i4 

643 

821 

828 

835 

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848 

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862 

868 

875 

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644 

889 

895 

902 

909 

916 

922 

929 

936 

943 

949 

645 

956 

g63 

969 

976 

983 

990 

996 

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646 

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057 

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070 

077 

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647 

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097 

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1  1  1 

117 

124 

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648 

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1  64 

171 

178 

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191 

198 

204 

211 

218 

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224 

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245 

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258 

265 

271 

278 

285 

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291 

298 

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325 

33i 

338 

345 

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N 

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1 

2 

3 

4 

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N 

0 

1 

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4 

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650 

81  291 

298 

305 

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3i8 

325 

33i 

338 

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65i 

358 

365 

37i 

378 

385 

39i 

398 

405 

4n 

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662 

425 

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438 

445 

45i 

458 

465 

47i 

478 

485 

653 

491 

498 

505 

5n 

5i8 

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538 

544 

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654 

558 

564 

57i 

578 

584 

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598 

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6i7 

655 

624 

63i 

637 

644 

65i 

657 

664 

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656 

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697 

7o4 

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7i7 

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743 

750 

657 

757 

763 

77° 

776 

783 

79° 

796 

8o3 

809 

816 

658 

823 

829 

836 

842 

849 

856 

862 

869 

875 

882 

659 

889 

895 

902 

908 

9i5 

921 

928 

935 

94  1 

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660 

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961 

968 

974 

981 

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994 

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661 

82  020 

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060 

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086 

092 

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112 

119 

125 

132 

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663 

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1  58 

1  64 

171 

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1  84 

191 

197 

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210 

664 

217 

223 

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236 

243 

249 

256 

263 

269 

276 

665 

282 

289 

295 

302 

3o8 

3i5 

321 

328 

334 

34i 

666 

347 

354 

36o 

367 

373 

38o 

387 

393 

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667 

4i3 

419 

426 

432 

439 

445 

452 

458 

465 

47i 

668 

478 

484 

491 

497 

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53o 

536 

669 

543 

549 

556 

562 

569 

575 

582 

588 

595 

601 

670 

607 

6i4 

620 

627 

633 

64o 

646 

653 

659 

666 

671 

672 

679 

685 

692 

698 

7°5 

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718 

724 

73o 

672 

737 

743 

750 

756 

763 

769 

776 

782 

789 

795 

673 

802 

808 

8i4 

821 

827 

834 

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847 

853 

860 

674 

866 

872 

879 

885 

892 

898 

9°5 

911 

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675 

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937 

943 

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963 

969 

975 

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078 

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123 

129 

1  36 

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161 

1  68 

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181 

679 

187 

198 

200 

206 

2l3 

219 

225 

232 

238 

245 

680 

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257 

264 

270 

276 

283 

289 

296 

302 

3o8 

N 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

PP      7 

6 

i   0.7               i 

0.6 

2     1.4                      2 

1.2 

3   2.1                3 

1.8 

4   2.8              4 

2.4 

5   3.5              5 

3.o 

6   4.2              6 

3.6 

7   4.9               7 

4.2 

8   5.6               8 

4.8 

Q   6.3               9 

5.4 

18 


68O— 72O 


N 

0 

1 

2 

3 

4 

5 

6 

7    8 

9 

680 

83  261 

257 

264 

270 

276 

283 

289 

296 

302 

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68  1 

315 

321 

327 

334 

34o 

347 

353 

359 

366 

372 

682 

378 

385 

391 

398 

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423 

429 

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683 

442 

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46i 

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474 

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487 

493 

499 

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5o6 

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53i 

537 

544 

55o 

556 

563 

685 

569 

575 

582 

588 

594 

601 

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620 

626 

686 

632 

639 

645 

65i 

658 

664 

67o 

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683 

689 

687 

696 

702 

708 

7i5 

721 

727 

734 

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688 

759 

765 

77i 

778 

784 

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797 

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809 

816 

689 

822 

828 

835 

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847 

853 

860 

866 

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879 

690 

885 

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897 

904 

910 

916 

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692 

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061 

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693 

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080 

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092 

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105 

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123 

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694 

1  36 

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1  48 

155 

161 

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180 

1  86 

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695 

198 

205 

211 

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223 

230 

236 

242 

248 

255 

696 

261 

267 

273 

280 

286  292 

298 

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323 

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336 

342 

348  354 

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367 

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379 

698 

386 

392 

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429 

435 

442 

699 

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466 

473 

479 

485 

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700 

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528 

535 

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547 

553 

559 

566 

701 

572 

578 

584 

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597 

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609 

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702 

634 

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646 

652 

658 

665 

67i 

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7i4 

720 

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733 

739 

745 

75i 

704 

757 

763 

•770 

776 

782 

788 

794 

800 

807 

8i3 

7o5 

819 

825 

83i 

837 

844 

850 

856 

862 

868 

874 

706 

880 

887 

893 

899 

9o5 

911 

9i7 

924 

93o 

936 

707 

942 

948 

954 

960 

967 

973 

979 

985 

991 

997 

708 

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009 

016 

022 

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052 

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709 

065 

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089 

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1  20 

710 

126 

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169 

175 

181 

711 

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199 

205 

211 

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224 

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236 

242 

712 

248 

254 

260 

266 

272 

278 

285 

291 

297 

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309 

3i5 

321 

327 

333 

339 

345 

352 

358 

364 

7i4 

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376 

382 

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394 

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412 

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437 

443 

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46  1 

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473 

479 

485 

716 

491 

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522 

528 

534 

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546 

717 

552 

558 

564 

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576 

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588 

594 

600 

606 

718 

612 

618 

625 

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637 

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N 

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1 

2 

3 

4 

5 

6 

7 

8 

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N 

0 

1 

2 

3 

4 

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7 

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720 

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800 

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812 

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842 

848 

722 

854 

860 

866 

872 

878 

884 

890 

896 

902 

908 

723 

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920 

926 

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938 

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950 

956 

962 

968 

724 

974 

980 

986 

992 

998 

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725 

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112 

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171 

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201 

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273 

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332 

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356 

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74  1 

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759 

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782 

788 

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800 

738 

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859 

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864 

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888 

894 

900 

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911 

917 

740 

923 

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999 

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105 

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116 

122 

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1  34 

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1  69, 

175 

181 

186 

192 

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216 

221 

227 

233 

239 

245 

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256 

262 

268 

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274 

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286 

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315 

320 

326 

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379 

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1 

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1.2                      2 

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N 

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668 

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685 

691 

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111 

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800 

806 

812 

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823 

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852 

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864 

869 

875 

881 

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892 

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767 

910 

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247 

763 

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270 

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326 

332 

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349 

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765 

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372 

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383 

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781 

265 

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276 

282 

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326 

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3o8 
354 
399 

444 
489 
534 

5-79 
623 
668 

3i3 
358 
4o3 

448 
493 
538 

583 
628 
673 

677 

682 

686 

691 

695 

700 

704 

709 

7i3 

717 

722 
76.7 

in 

856 
900 
945 
989 
99  o34 
078 

726 
771 
816 

860 

9°5 
949 

994 
o38 
o83 

73i 
776 
820 

865 
909 
954 

998 
o43 
087 

735 
780 
825 

869 
914 
958 

*oo3 
047 
092 

74o 
784 
829 

874 
918 
963 

*oo7 

052 

096 

744 
789 
834 

878 
923 
967 

*OI2 

o56 

IOO 

749 
793 

838 

883 
927 
972 

*oi6 
06  1 
105 

753 
798 

843 

887 
932 
976 

*02I 
065 
109 

758 
802 
847 

892 
936 
981 

*025 

069 
n4 

762 
807 
85  1 

896 
94i 

985 

*029 
074 

118 

123 

127 

i3i 

i36 

i4<> 

i45 

1  49 

1  54 

i58 

162 

981 
982 
983 

984 
985 
986 

987 
988 
989 

990 

991 

992 
993 

994 
995 
996 

997 

998 

999 
1000 

167 

21  I 

255 

3oo 
344 

388 

432 
476 

52O 

171 
216 
260 

3o4 
348 
392 

436 

48o 
524 

176 
220 
264 

3o8 
352 

396 

44  1 
484 
528 

180 
224 
269 

3i3 
357 
4oi 

44s 
489 
533 

185 

229 
273 

3i7 
36i 

4o5 

449 
493 
537 

189 
233 
277 

322 

366 
4io 

454 
498 
542 

I93 

238 
282 

326 
37o 

4i4 

458 

502 

546 

198 
242 
286 

33o 

374 
419 

463 
5o6 
55o 

202 

247 
291 

335 
379 

423 

467 
5n 
555 

207 

25l 
295 

339 
383 
427 

47i 
5i5 
559 

564 

568 

572 

577 

58i 

585 

590 

594 

599 

6o3 

607 
65i 
695 

739 

782 
826 

870 
9i3 
957 

612 
656 
699 

743 
787 
83o 

874 
917 
961 

616 
660 
704 

747 
791 
835 

878 
922 
965 

621 

664 
708 

752 
795 
839 

883 
926 
97° 

625 
669 
712 

756 
800 

843 

887 
93o 

974 

629 
673 
717 

760 

8o4 
848 

891 

935 
978 

634 
677 
721 

765 

808 
852 

896 

939 

983 

638 
682 
726 

769 
8:3 
856 

9oo 

944 
987 

642 
686 
73o 

774 
817 
861 

9o4 
948 
99i 

647 
691 

734 

778 
822 
865 

909 

952 
996 

00  OOO 

oo4 

009 

oi3 

017 

022 

026 

o3o 

o3§ 

o39 

X 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

27 


TABLE  II 

FIVE -PL  ACE     LOGARITHMS 

OF    THE 

TRIGONOMETRIC    FUNCTIONS 

TO  EVERY  MINUTE 


0°. 


L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L. 

Cos. 

0 

— 

— 

— 

0  .  OO  OOO 

60 

I 

6.46  373 

30103 

6.46  373 

3.53627 

O.OO  OOO 

59 

2 

3 

6.94085 

17609 

6.76476 
6.94085 

30103 
17609 

3.23  524 
3.o59i5 

O.OO  OOO 
0.00  000 

58 
57 

4 

7.06679 

12494 
9691 

7.06  579 

12494 

nfinr 

2.9 

3421 

O.OO  OOO 

56 

5 

7.16  270 

7018 

7.16  270 

909 

2.8 

373o 

O.OO  OOO 

55 

6 

7.24  188 

7910 

7.24  188 

791? 

2.75  812 

O.OO  OOO 

54 

6694 

6694 

7 

7.30882 

7.30882 

c8oo 

2.69  1  1  8 

O.OO  OOO 

53 

8 

7.36682 

7.36682 

2.633:8 

O.OO  OOO 

52 

9 

7-4i  797 

5"5 

7-4i  797 

5"5 

2.58  2o3 

O.OO  OOO 

5i 

10 

7.46373 

7.46  373 

457° 

2.53  627 

O.OO  OOO 

50 

1  1 

7.5o  5i2 

4J39 

7.5o  5i2 

4139 

2.49488 

O.OO  OOO 

49 

12 

i3 

7.54  291 
7.57  767 

3476 

7.54  291 
7.57767 

3476 

2.45  709 
2.42  233 

O.OO  OOO 
O.OO  OOO 

48 
47 

i4 
i5 

7.60985 
7.63  982 

3218 
2997 

7.60  986 
7.63  982 

3219 
2996 

2.  Sg  Ol4 

2.36  018 

0.00  000 
O.OO  OOO 

46 

45 

16 

7.66784 

7.66785 

2803 

2.332:5 

O.OO  OOO 

44 

2633 

263 

1 

17 

7-69417 

2483 

7.69418 

2482 

2.3o  582 

9-99999 

43 

18 

7.71  900 

7-74248 

2348 

7.71  900 

7.74248 

2348 

2.28  :oo 

2.25  752 

9-99999 
9-99999 

42 

4i 

20 

7.76475 

2227 

7.76476 

2.23  524 

9-99999 

40 

21 

7-7* 

J  594 

7.78595 

2.2:  405 

9-99999 

39 

22 

7.80615 

7.80  6i5 

2.:9385 

9-99999 

38 

23 

7.82545 

1930 

7.82546 

*93 

2.:7  454 

9-99999 

37 

1848 

184 

1 

24 

7.84393 

7-84394 

2.:  5  606 

9  -.99  999 

36 

25 

7.8( 

5  166 

7.86167 

2.:3833 

9-99999 

35 

26 

7.87870 

1704 

7.87871 

1704 

2.:2  :29 

9-99999 

34 

1639 

i63( 

1 

27 
28 

7«89  509 
7.91  088 

1579 

7.89  5io 
7.91  089 

1579 

2.  :o  49o 
2.08  9:  i 

9-99999 
9-99999 

33 

32 

29 

7.92  612 

1524 

7.92  6i3 

152- 

\ 

2.07  387 

9-99  99s 

3: 

30 

7.94  o84 

7.94  086 

1 

2.o5  9:4 

9-99998 

30 

L.  Cos. 

d. 

L.  Cotg.       d. 

L.  Tang. 

L.  Sin. 

' 

89°  3O  . 

PP 

9691 

4576 

2997 

2483 

2119 

I848 

1704 

1579 

1472 

.1 

969 

458 

300 

.x 

248 

212 

185 

.! 

170 

158 

*47 

.2 

1938 

.2 

497" 

424 

37° 

.2 

341 

316 

294 

•3 

2907 

1372 

899 

•3 

745 

636 

554 

•3 

5" 

474 

442 

•4 

3876 

1830 

"99 

•4 

993 

848 

739 

•4 

682 

632 

589 

5 

4846 

2288 

1498 

•5 

1242 

1060 

924 

•5 

8.S2 

789 

736 

.6 

2646 

1798 

.6 

i49p 

I27I 

1109 

.6 

IO22 

947 

883 

•7 

6784 

3203 

2098 

•  7 

1738 

M83 

1294 

.7 

"93 

1105 

1030 

.8 

7753 

3661 

2398 

.8 

1986 

l695 

1478 

.8 

1263 

1178 

4118     2697 

1663 

1421 

1325 

O°  3O . 


L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L. 

Cos. 

30 

7  .  94  o84 

7.94  086 

2.o5  914 

9.99998 

30 

3i 

32 

7.95  5o8 
7.96  887 

1424 
1379 

7.96  5io 
7.96  889 

1424 

1379 

2.04  490 

2.O3  III 

9.99998 
9.99998 

29 

28 

33 

7.98  223 

133° 

7-9* 

1  225 

2.01  775 

9.99998 

27 

1297 

129; 

34 

7.99  520 

7.99  522 

1259 

2.OO  478 

9.99998 

26 

35 
36 

8.00  779 
8.  02  002 

1223 
1190 

8.00  781 

8.02  004 

1223 
1190 

I.992I9 
1.97996 

9.99998 
9.99998 

25 
24 

37 

8.o3  192 

1158 

8.o3  194 

1159 

I  .96  806 

9-99  997 

23 

38 

8.o4  35o 

1128 

8.04353 

i  .96  647 

9-99997 

22 

39 

8.o5 

478 

8.o548i 

i  .  94  5  1  9 

9-99997 

21 

40 

8.06  578 

8.06  58i 

1.93419 

9-99997 

20 

4i 

8.07  650 

1046 

8.07  653 

1047 

1.92  347 

9-99997 

I9 

42 

8.08  696 

8.08  700 

i  .91  3oo 

9-99997 

18 

43 

8.09  718 

8.09  722 

1022 

i  .90  278 

9-99997 

17 

999 

998 

44 

8.  10  717 

976 

8.  10  720 

i  .89  280 

9.99996 

16 

45 

8.  ii 

693 

8.  ii 

696 

1.88  3o4 

9.99996 

i5 

46 

8.12 

647 

954 

8.12  65  1 

955 

1.87  349 

9.99996 

i4 

47 

8.i3  58i 

934 

8.i3  585 

934 

1.86415 

9.99996 

i3 

48 

8.i4 

495 

8.  1  4  500 

i.85  5oo 

9.99996 

12 

49 

8.16391 

896 

8.i5395 

895 

0_0 

i.84  605 

9.99996 

II 

50 

8.16268 

877 

8.16273 

i.  83  727 

9.99995 

10 

5i 

8.17  128 

840 

8.17  i33 

843 

i  .82  867 

9.99996 

9 

52 

8.17  971 

8.17976 

828 

i  .82  024 

9-99  995 

8 

53 

8.18  798 

27 
812 

8.18  8o4 

812 

i  .81  196 

9-99995 

7 

54 
55 

'  8.19  610 
8.  20  407 

797 

8.19  616 
8.2o4i3 

797 

_0_ 

i.  80  384 
1.79687 

9-99995 
9.99994 

6 
5 

56 

8.21 

189 

782 

8.21  196 

702 

1.78  805 

9.99994 

4 

57 

8.21 

968 

769 

8.21  964 

769 

7rA 

1.78036 

9.99994 

3 

58 

8.22  713 

755 

8.22  720 

1.77  280 

9.99994 

2 

59 

8.23456 

743 

8.23462 

742 

1.76  538 

9-99  994 

I 

60 

8.24  186 

730 

8.24  192 

73° 

1.75808 

9.99993 

0 

L.  Cos.        d. 

L.  Cotg. 

d. 

L,  Tang. 

L. 

Sin. 

' 

89°. 

PP   1379 

1223 

IIOO 

999 

914 

860 

812 

769 

730 

r 

138 

122 

no 

.1 

IOO 

9* 

86 

i 

81 

77 

73 

.2 

276 

245 

220 

.2 

200 

183 

172 

2 

162 

146 

•3 

414 

367 

330 

•3 

300 

274 

258 

3 

244 

231 

219 

•  4 

552 

489 

440 

•4 

400 

366 

344 

4 

325 

308 

292 

690 

612 

500 

457 

430 

5 

406 

tf-5 

365 

.6 

827 

734 

660 

.6 

599 

548 

6 

487 

461 

438 

•7 

96s 

856 

770 

7 

699 

640 

602 

7 

568 

538 

5" 

.8 

1103 

978 

880 

.8 

799 

73i 

688 

8 

650 

615 

584 

uoi         990 

9 

899        823 

774 

9        731 

692 

657 

3 1 


1°. 


, 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L. 

Cos. 

0 

8.24  186 

8.24  192 

i.75  808 

9.99993 

60 

I 

8.24903 

717 
706 

8.24  910 

710 
706 

i  .75  090 

9.99  993 

59 

2 

8.25  609 

8.25  616 

1.74384 

9.99  993 

58 

3 

8.26  3o4 

095 

684 

8.26  3i2 

2J 

1.73688 

9.99993 

5? 

4 

8.26  988 

673 

8.26  996 

671 

1.73  oo4 

9.99992 

56 

5 

8.27  661 

8.27  669 

i  .72  33i 

9.99  992 

55 

6 

8.28  324 

663 

8.28  332 

663 

1.71  668 

9.99992 

54 

7 

8.28977 

653 

8.28  986 

64? 

1.71  oi4 

9.99992 

53 

8 

8  .29  621 

8  .29  629 

1.70  37i 

9.99992 

52 

9 

8.3o  255 

°34 

8.3o  263 

634 

i.69737 

9-99991 

5i 

10 

8,3o  879 

024 
616 

8.3o888 

625 

i  .69  112 

9-99  991 

50 

n 

8.3i 

495 

608 

8.3i  5o5 

17 
607 

i.68495 

9-99  991 

49 

12 

8.32  io3 

8.32   112 

1.67888 

9.99990 

48 

:3 

8.32  702 

599 

8.32  711 

599 

1.67  289 

9-9999° 

47 

14 

8.33  292 

59° 

8.33  3o2 

^ 

1.66698 

9.99990 

46 

i5 

8.33875 

8.33  886 

5  4 

i  .66  1  14 

9.99990 

45 

16 

8.3445o 

568 

8.3446r 

575 

568 

i.65539 

9.99989 

44 

17 

8.35oi8 

560 

8.35  029 

i  .64  971 

9.99989 

43 

18 

8.35  578 

8.35  590 

5 

i  .64  4io 

9.99989 

42 

19 

8.36  i3i 

553 

8.36  i43 

553 

i.63857 

9-99  989 

4i 

20 

8.  36  678 

8.  36  689 

M° 

i.63  3n 

9.99  988 

40 

21 

8.37  217 

533 

8.37  229 

54° 

i  .62  771 

9-99988 

39 

22 

8.37750 

526 

8.37  762 

1.62  238 

9.99988 

38 

23 

8.38  276 

520 

8.38  289 

527 
520 

i  .61  711 

9.99987 

3? 

24 

8.38  796 

514 

8.38  809 

i  .61  191 

9.99987 

36 

25 

8.39  3io 

8.39  323 

i  .60  677 

9.99987 

35 

26 

8.  398i8 

SOB 

8.39832 

s°9 

i.  60  168 

9.99986 

34 

27 

8.4o  320 

502 
406 

8.4o  334 

502 

406 

1.59666 

9.99986 

33 

28 

8.4o  816 

8.4o83o 

i  .5 

9  170 

9.99  986 

32 

29 

8.4i  307 

491 

8.4i  32i 

491 

i.5 

8679 

9.99  985 

3i 

30 

8.4i  792 

485 

8.4i  807 

486 

i.58  i93 

9.99985 

30 

L.  Cos. 

d. 

L.  Cotg.      d. 

L.  Tang. 

L. 

Sin. 

' 

88°  30  . 

PP 

706 

663 

634 

599 

575 

553 

533 

SM 

496 

.! 

70.6 

66.3 

63.4 

.1 

59-9 

57-5 

55-3 

.! 

53-3 

51-4 

49-6 

.2 

141.2 

132.6 

126.8 

.2 

119.8 

115.0 

1  10.  6 

.2 

106.6 

102.8 

99.2 

•3 

211.  8 

198.9 

190.2 

•3 

179.7 

172-5 

165.9 

•3 

159-9 

I54-2 

148.8 

•4 

282.4 

265.2 

253-6 

•4 

239.6 

230.0 

221.2 

•4 

213.2 

205.6 

198.4 

•  5 

353-° 

33I-5 

317-° 

t  e 

299-5 

287.5 

276.5 

•  5 

266.5 

257.0 

248.0 

.6 

423.6 

397-8 

380.4 

6 

359-4 

345-0 

33^.8 

.6 

319.8 

308.4 

297.6 

•  7 

494.2 

464.1 

443-8 

•7 

4r9-3 

402.5 

387.1 

.7 

373-1 

359-8 

347-2 

.8 

564.8 

53°-  4 

507.2 

.8 

479-2 

460.0 

442-4 

.8 

426.4 

411.2 

396.8 

.9     635.4 

596.7  '  570.6 

539.1     5T7-5 

497  -J 

•9     479-7 

446.4 

32 


> 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L. 

Cos. 

30 

8.4i  792 

8.4i  807 

i  .5s  i93 

9.99985 

30 

3i 

32 

8.42  272 
8.42  746 

480 

474 

8.42  287 
8.42  762 

475 

i.577i3 
1.57  238 

9.99985 
9.99  984 

29 

28 

33 

8.43  216 

470 
464 

8.43  232 

470 
464 

i.56  768 

9.99984 

27 

34 
35 

8.43  680 
8.44  i39 

459 

8.  43  696 
8.44  i56 

460 

i.56  3o4 
i.55  844 

9.99984 
9.99  983 

26 
25 

36 

8  .44  594 

455 

8.446u 

455 

i.55389 

9.99983 

24 

37 

38 

8.45  o44 
8.45489 

45<> 
445 

8.45o6i 
8.45  507 

450 
446 

1.54939 
1.54493 

9.99983 
9.99982 

23 
22 

39 

8.45  930 

441 

8.45  948 

44' 

i.54o52 

9.99982 

21 

40 

8.46  366 

430 

8.46385 

437 

i.536i5 

9.99982 

20 

4i 

8.46  799 

433 

8.46817 

432 

428 

i.53  i83 

9.99  981 

19 

42 

8.47  226 

8.47  245 

1.52  755 

9.99981 

18 

43 

8.47650 

424 

8.47  669 

424 

i.52  33i 

9.99981 

'7 

419 

4* 

44 
45 

8.48  069 
8.48485 

416 

8.48  089 
8.485o5 

416 

i  .  5i  911 
i.5i  495 

9.99980 
9.99980 

16 
i5 

46 

8.48896 

411 

408 

8.4* 

J  917 

412 

408 

j.5i  o83 

9.99979 

i4 

47 

8.49  3o4 

8.49  325 

i    5o  675 

9.99979 

i3 

48 

8.49  708 

404 

8.49  729 

i  .5o  271 

9.99979 

12 

49 

8.5o  1  08 

400 

8.5o  i3o 

401 

i  .49  870 

9.99  978 

I  I 

50 

8.5o  5o4 

39° 

8.5o  527 

397 

i  .49  473 

9.99978 

10 

5i 

8.5o  897 

393 

8.5o  920 

393 

i  .49  080 

9.99977 

9 

52 

8.5i 

287 

-Of: 

8.5i  3io 

-Of. 

i  .48  690 

9.99977 

8 

53 

8.5i 

673 

330 

382 

8.5i  696 

300 
383 

i.48  3o4 

9  99977 

7 

54 

8.52  o55 

8.52  079 

180 

i  .47  921 

9.99976 

6 

55 

8.52434 

8.52459 

i.4754i 

9.99976 

5 

56 

8.52  810 

376 

8.52  835 

376 

1.47  i65 

9.99975 

4 

373 

37: 

57 

8.53  i83 

060 

8.53  208 

i  .46  792 

9-99975 

3 

58 

8.53  552 

8.53  578 

i  .46  422 

9.99974 

2 

59 

8.53  919 

367 

8.  53  945 

367 

,6, 

i.46o55 

9.99974 

I 

60 

8.54  282 

8.54  3o8 

i  .45  692 

9-99  974 

0 

L.  Cos.        d. 

L.  Cotg. 

d. 

L.  Tang. 

L. 

Sin. 

! 

88°. 

PP 

470 

455 

441 

424 

4o8 

396 

386 

376 

367 

.! 

47.0 

45-5 

44.1 

.1 

42.4 

40.8 

39-6               -i 

38.6 

376 

36.7 

.  2 

94.0 

91  o 

88.2 

.2 

84.8 

81.6 

79.2               .2 

77-2 

752 

73-4 

•3 

141.0 

136-5 

132-3 

•3 

127.2 

122.4 

118.8               .3 

1128 

1  10.  1 

•4 

188.0 

182.0 

176.4 

•4 

169.6 

163.2 

158.4               .4 

154-4 

1504 

146.8 

•  5 

235-0 

227.5 

220.5 

•  5 

2I2.O 

204.0 

198.0               .5 

193.0 

1880 

183-5 

6 

282.0 

273.0 

264.6 

.6 

254-4 

244.8 

237-6              -6 

231.6 

225.6 

220.2 

.7 

329.0 

318.5 

308.7 

•  7 

296.8 

285.6 

277.2               .7 

270.2 

263.2 

256.9 

.8 

376.0 

364.0 

352.8 

.8 

339-2 

326.4 

316.8               .8 

308.8 

300.8 

293-6 

•9      423- 

409.  5     396.  9 

381.6     367.2 

356-4                -9      347-4 

338.4 

33°-3 

33 


2°. 


/ 

L. 

Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L. 

Cos. 

0 

8.54282 

760 

8.54  3o8 

i  .45  692 

9.99974 

60 

I 

8.54642 

357 

8.54669 

358 

i.45  33i 

9.99  973 

59 

2 

8.54  999 

8.55  027 

i.44  973 

9.99973 

58 

3 

8.55  354 

355 

8.55  382 

355 

i.446i8 

9.99  972 

57 

4 

8.557o5 

8.55734 

352 

1.44266 

9.99  972 

56 

5 

8.56o54 

8.56o83 

349 

i  .43  917 

9.99  971 

55 

6 

8.56  4oo 

346 

8.56429 

346 

i.43  57i 

9.99971 

54 

7 

8.56  743 

343 

8.56  773 

344 

i.43  227 

9.99  97o 

53 

8 

8.67  o84 

8.57  n4 

1.42  886 

9.99  9-70 

52 

9 

8.57  421 

337 

8.57452 

338 

1.42  548 

9-99  969 

5i 

10 

8.57757 

33° 

8.57788 

336 

I  .42  212 

9-99  969 

50 

1  1 

8.58  089 

8.58  121 

333 
33° 

I  .4l    879 

9.99968 

49 

12 

i3 

8.584i9 
8.58747 

328 

8.5845i 
8.58779 

328 

i  .41  549 

I  .4l   221 

9.99968 
9.99  967 

48 
47 

i4 
i5 

8.59  072 
8.59395 

325 
323 

8.59  io5 
8.69428 

326 
323 

i.4o  895 
1  .40  672 

9.9996-7 
9.9996-7 

46 

45 

16 

8.69  716 

318 

8.59  749 

321 

i  .4o  261 

9.99966 

44 

17 
18 

8.6oo33 
8.60  349 

3i6 

8.60068 
8.60  384 

316 

1.39  932 
1.39  616 

9.99966 
9.99965 

43 

42 

'9 

8.60  662 

3*3 

8.60  698 

3i 

| 

1.39  3o2 

9.99  964 

4i 

20 

8.60  973 

3°9 

8.  6  r  009 

3" 

i.38  991 

9.99  964 

40 

21 

8.61  282 

307 

8.61  3i9 

310 

i.  3868i 

9.99963 

39 

22 

8.61  589 

8.61  626 

i.38  374 

9.99963 

38 

23 

8.61  894 

8.61  931 

3°5 

i.38  069 

9.99962 

37 

302 

3° 

1 

24 

8.62  196 

301 

8.62234 

i.  37766 

9.99962 

36 

25 

8.62  497 

8.62  535 

1.37465 

9.99961 

35 

26 

8.62795 

298 

8.62  834 

299 

1.37  166 

9.99961 

34 

27 
28 

8.63  091 
8.63  385 

296 
294 

8.63  i3i 
8.63426 

297 
295 

i.36  869 
i.36574 

9.99960 
9.99960 

33 

32 

29 

8.  63  678 

293 

8.63  718 

292 

1.36282 

9.99  959 

3i 

30 

8.  63  968 

290 

8.64  009 

291 

i.35  991 

9.99959 

30 

L. 

Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L. 

Sin. 

' 

87°  3O'. 

PP 

360 

350 

34° 

33° 

320 

310 

300 

290 

285 

.1 

36 

35 

34 

.1 

33 

32 

31                 .1 

30 

29 

28.5 

.2 

72 

7° 

68 

.2 

66 

64 

62                        .2 

60 

58 

57-° 

•3 

1  08 

105 

1  02 

•3 

99 

96 

93                 -3 

90 

87 

85-5 

•4 

144 

140 

136 

•4 

132 

128 

124                 .4 

120 

116 

114.0 

180 

170 

•65 

160 

ISO 

145 

M2.5 

.6 

216 

2IO 

204 

.6 

198 

192 

1  86                 .6 

1  80 

171.0 

•7 

252 

245 

238 

•  7 

231 

224 

217                 .7 

2IO 

203 

199-5 

.8 

288 

280 

272 

.8 

264 

256 

248 

240 

232 

228.0 

.0 

324 

315 

306 

2Q7 

288 

279                 .9        270 

161 

256.5 

2°  3O  . 


/ 

L. 

Sin. 

d. 

L.  Tang. 

d. 

L. 

Cotg. 

L. 

Cos. 

30 

8.63  968 

8  .64  009 

i  .35  991 

9.99  959 

30 

3i 

32 

8.64256 
8.64543 

287 

8.64  298 
8.64  585 

289 
287 

i  .35  702 
i.35  415 

9.99958 

29 

28 

33 

8.6 

4827 

284 
283 

8.64  870 

285 
284 

i.35  i3o 

9.99957 

27 

34 

8.65  no 

8.65  i54 

1.34846 

9.99  956 

26 

35 

8.6539i 

8.65435 

1.34565 

9.99956 

25 

36 

8.  65  670 

279 

8.  65  715 

280 

1.34285 

9.99955 

24 

37 
38 

8.65  947 
8.66223 

277 
276 

8.65  993 
8.66  269 

278 
276 

i  .34  007 
i.33  73i 

9.99955 
9.99954 

23 
22 

39 

8.66  497 

274 

8.66543 

274 

1.33457 

9-99954 

21 

40 

8.66  769 

272 

8.66816 

273 

i.33  1  84 

9-99  953 

20 

4i 

42 

8.67  039 
8.673o8 

270 
269 

8.67  087 
8.  67  356 

271 
269 

i.32  913 
i.32  644 

9.99952 
9.99952 

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18 

43 

8.67575 

267 
266 

8.67624 

266 

i.32376 

9.99961 

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44 
45 

8.67841 
8.68  io4 

263 

8.67  890 
8.68  1  54 

264 

i  .32  no 
i.3i  846 

9.99961 
9.99950 

16 

i5 

46 

8.68  367 

263 
260 

8.684i7 

263 
261 

i.3i  583 

9.99949 

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47 

8.68627 

8.68678 

260 

i.: 

»I    322 

9.99949 

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48 

8.68886 

259 

8.68  938 

1.2 

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9.99948 

12 

49 

8.69  1  44 

258 

8.69  196 

258 

i.3o  8o4 

9.99948 

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50 

8.69  4oo 

256 

8.69453 

257 

i.3o547 

9.99  947 

10 

5i 

8.  69  654 

254 

8.69  708 

255 

i  .3o  292 

9.99946 

9 

52 

8.69  907 

8.69  962 

i.3oo38 

Q.  99946 

8 

63 

8.70  i59 

252 

8.70  214 

252 

1.29  786 

9.99945 

7 

250 

25 

54 

8.70  409 

8.70465 

i.  29  535 

9.99944 

6 

55 

8.70658 

8.70  714 

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9.99944 

5 

56 

8.70  905 

247 

246 

8.70  962 

240 
246 

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9.99943 

4 

57 

8.71  i5i 

8.71  208 

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9.99942 

3 

58 

8.71  395 

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9.99942 

2 

59 

8.71  638 

243 

8.7i  697 

244 

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9.99941 

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60 

8.71  880 

8.71  940 

i  .28  060 

9-9994o 

0 

L. 

Cos. 

d. 

L.  Cotg. 

d.      L.  Tang. 

L. 

Sin. 

f 

87°. 

PP 

280 

275 

270 

265 

260 

255 

250 

245 

240 

.1 

28.0 

27-5 

27.0 

.1 

26.5 

26.0 

25-5 

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25.0 

24-5 

24.0 

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56.0 

55-0 

54-o 

.2 

53-o 

52.0 

51.0 

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50.0 

49-° 

48.0 

•3 

84.0 

82.5 

81.0 

•3 

79-5 

78.o 

76.5 

•3 

75-0 

73-5 

72.0 

•  4 

112.  0 

1  10.0 

108.0 

•4 

106.0 

104.0 

102.0 

•4 

100.0 

98.0 

96.0 

t  e 

140.0 

137-5 

i35-o 

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132-5 

130.0 

I27-5 

•5 

125.0 

122.5 

12O.O 

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168.0 

165.0 

162.0 

.6 

159.0 

156.0 

i53-o 

.6 

150.0 

147.0 

144.0 

.7 

196.0 

192.5 

189.0 

•7 

185-5 

182.0 

178-5 

•7 

175-0 

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168.0 

.8 

224.0 

22O.O 

216.0 

.8 

2I2.O       2O8.O 

204.0 

.8 

200.0 

196.0 

192.0 

•9 

247-5 

243.0 

•9 

238.5       234.0 

.9     225.0 

216.0 

35 


( 

L. 

Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L. 

Cos. 

0 

8.71  880 

340 
239 
238 
237 
235 
234 
232 
232 
230 
229 

228 
226 
226 
224 
223 

222 
2  2O 
2  2O 
2I9 
2I7 

216 
216 

2I4 
213 
212 
211 
210 
209 
208 
208 

8.71  940 

241 
239 
239 
237 

236 

234 
234 
232 
231 
229 

229 
227 
226 
225 
224 

222 
222 
220 
2I9 
2I9 

217 

216 

215 
214 

213 

211 
211 

210 
209 
208 

i  .28  060 

9  .99  940 

60 

I 

2 

3 

4 
5 
6 

8 
9 

8.72   I2O 

8.  72359 

8.72  597 

8.72  834 
8.73069 
8.73  3o3 

8.73  535 
8.73  767 
8.73  997 

8.72  181 
8.72  420 
8.72  659 

8.72  896 
8.73i32 
8.73366 

8.73  600 
8.73832 
8  .  74  06-3 

i  .27  819 
1.27  58o 
1.27  34  i 

1.27  io4 
1.26868 
1.26634 

i  .26  4oo 
1.26  168 
i  .25  937 

9.99  940 
9.99939 
9.99938 

9.99938 
9.99937 
9.99936 

9.99936 
9.99935 
9.99934 

59 
58 
57 

56 
55 
54 
53 

52 

5i 

10 

8.74  226 

8.74  292 

i.25  708 

9.99934 

50 

ii 

12 

i3 

i4 
i5 
16 

17 

18 

'9 

8.74454 
8.74  680 
8.74  906 

8.75  i3o 
8.  75353 
8.75575 

8.75795 
8.76oi5 
8.76234 

8.74  52i 
8.  74748 
8.74974 

8.75  199 
8.75  423 
8.75  645 

8.75867 
8.76087 
8.76  3o6 

i.25  479 

I  .25  252 
I  .25  026 

i  .24  801 
1.24  577 

1.24355 

1.24  i33 
i.23  913 
i.23  694 

9.99933 
9.99932 
9.99932 

9.99931 
9.99930 
9.99929 

9.99929 
9.99928 
9.99  927 

49 
48 

47 
46 
45 
44 

43 

42 

4r 

20 

8.7645i 

8.76  525 

I  .23  475 

9.99926 

40 

21 
22 
23 

24 
25 

26 

27 
28 
29 

8.76667 
8.  76  883 
8.77097 

8.77  3io 

8.77  522 
8.77943 

8.78  i52 
8.78  36o 

8.76  742 
8.76958 
8.77173 

8.77387 
8.77  600 
8.77  811 

8.78  022 
8.78232 

8.7844i 

i.23  258 

I  .23  O42 
I  .22  827 

1.22  6l3 
I  .22  400 
I  .22   189 

I  .21   978 
I.  21   768 
I  .21   559 

9.99926 
9.99925 
9.99924 

9.99923 
9.99923 
9.99922 

9.99921 
9.99920 
9.99  920 

39 

38 

37 
36 
35 
34 
33 

32 

3i 

30 

8.78  568 

8.78  649 

i.  2  1  35  1 

9.99  919 

30 

L. 

Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L. 

Sin. 

' 

80 

°3O 

. 

PP 

238 

234 

229 

.2 

•3 
•4 

:78 

•9 

225 

220 

216 

2X2 

208 

204 

.2 

•3 

•4 

•  9 

23.8 
47.6 
71.4 

95-2 
119.0 
142.8 

166.6 
190.4 

23-4 
46.8 
70.2 

93-6 
117.0 

140.4 

163.8 

187-2 

22.9 

a; 

91.6 

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137-4 

160.3 
183.2 
206.  i 

22.5 

45-0 
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90.0 
112.5 
135-0 

157-5 
180.0 

52-5 

22.  0 
44.0 

66.0 

88.0 

IIO.O 

132.0 

176.0 

21.6                        .1 

43-2       '         .2 

64.8        .3 
86.4    1     .4 

108.0                 .5 
129.6      !         .6 

151-2            .7 

172.8                .8 

194.4                .9 

21.2 

42.4 
63.6 

84.8 

ro6.o 
127.2 

148.4 
[69.6 

90.8 

20.8 

41.6 
62.4 

83.2 
104.0 
124.8 

145.6 
166.4 

187.2 

20.4 
40.8 
61.2 

81.6 

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122-4 

142.8 
163.2 
183.6 

36 


3°  3O'. 


/ 

L.  Sin. 

d. 

L.  Tang.      d. 

L.  Cotg. 

L. 

Cos. 

30 

8.78  568 

8.78  649 

i  .21  35i 

9-99  9'9 

30 

3i 

8.78  774 

8.78  855 

206 
206 

1.21  145 

9.99918 

29 

32 

8.78979 

8.70.  061 

i  .20  939 

9.99917 

28 

33 

8.79  i83 

204 

8.79266 

205 

I  .20  734 

9.99917 

27 

203 

20< 

34 
35 

8.79  386 
8.79  588 

202 

8.79  4?o 
8.79673 

203 

i  .20  53o 

i  .20  327 

9.99916 
9.99915 

26 
25 

36 

8.79789 

8.79875 

I  .20  125 

9.99  9i4 

24 

20J 

37 

8.79990 

8.80076 

1.19  924 

9.99913 

23 

38 

8.80  189 

199 

8.80  277 

1  .19  723 

9.99913 

22 

39 

8.  80  388 

199 

8.80476 

199 

i  .  19  524 

9.99912 

21 

40 

8.  80  585 

197 

8.8o674 

198 

1.19  326 

9.99911 

20 

4i 

8.80782 

197 
1  06 

8.80  872 

198 
1  06 

1.19  128 

9.99910 

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42 

43 

8.80  978 
8.81  173 

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8.81  068 
8.81  264 

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1.  18  932 
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9.99909 
9.99909 

1  8 

17 

194 

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44 

8.81  367 

8.81  459 

1.  18  54i 

9.99908 

16 

45 

8.81  56o 

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8.81  653 

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i.i8347 

9.99907 

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8.81  752 

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8.81  846 

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1.  18  i54 

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8.81  944 

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48 

8.82  1  34 

8.82  23o 

1.17  770 

9.99904 

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49 

8.82  324 

190 

8.82  420 

190 

1.17  58o 

9.99904 

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50 

8.82  5i3 

8.82  610 

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9.999o3 

10 

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8.82  701 

8.82  799 

1  88 

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9.99902 

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r 

8.82888 

8.820,87 

18 

5 

1.17  oi3 

9.99901 

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53 

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1.16825 

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54 

8.83  261 

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8.8336i 

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1.16  639 

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8.83446 

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56 

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8.83732 

105 
184 

1.16268 

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57 

8.83  8i3 

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3 

58 

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8.84  177 

181 

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9-99  895 

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8.84  358 

8.84464 

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9.99  894 

0 

L. 

Cos.        d. 

L.  Cotg.       d 

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L. 

Sin. 

' 

86°. 

PP 

201 

198 

195 

192 

I89 

187 

185 

183 

181 

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60.3 

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74.0 

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97-5 

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96.0 

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91.5 

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8 

117.0 

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109.8 

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140.7 

138.6 

136-5 

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132.3 

130.9 

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129.5 

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126.7 

160.8 

158.4 

156.0 

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151.2 

149.6 

.8 

148.0 

146.4 

r44.8 

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180.9 

178.2     175.5 

168.3 

164.7     '62.9 

4°. 


/ 

L. 

Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L. 

Cos. 

0 

8.84358 

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8.84464 

1.  15536 

9.99  894 

60 

I 

8.  84539 

179 

8.84646 

1  80 

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9.99893 

r 

2 

8.8 

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8.84  826 

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9.99  892 

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3 

8.8 

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179 

8.85  006 

1  80 

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9.99891 

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4 

8.85  075 

178 

8.85  185 

179 

T,o 

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9.99  891 

56 

5 

8.85  252 

8.85363 

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9.99890 

55 

6 

8.85429 

177 

8.85  54o 

177 

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9.99889 

54 

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8.85605 

176 

8.85  717 

177 

176 

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53 

8 

8.85  780 

8.85893 

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9.99887 

52 

9 

8.85  955 

175 

8.86069 

176 

i.i393i 

9.99  886 

5i 

10 

8.86  128 

173 

8.86  243 

174 

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9.99  885 

50 

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8.86  3oi 

8.86417 

1.  13583 

9.99884 

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8.86474 

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1.  1  3  409 

9.99  883 

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8.86645 

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8.86763 

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i.i3237 

9.99  882 

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8.86816 

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8.87  106. 

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9.99  880 

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8.87  i56 

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8.87277 

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170 

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9.998-79 

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8.  87  325 
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8.  87  447 
8.87  616 

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8.87661 

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8.87  829 

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1.  12  047 

9.99  876 

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21 

8.87995 

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8.88  120 

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9.99  875 

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22 

8.88  161 

8.88  287 

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9.998-74 

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23 

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9.99873 

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8.88  490 

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8.88  618 

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8.8 

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9.99871 

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26 

8.88817 

103 

163 

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9.99870 

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8.88  980 

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8.89  in 

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1.10889 

9.99869 

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28 

8.8 

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8.89  274 

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9.99868 

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29 

8.89  3o4 

162 

8.89437 

163 

i.io563 

9.99867 

3i 

30 

8.89464 

8.89  598 

I.I0402 

9.99  866 

30 

L. 

Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L. 

Sin. 

9 

85°  30  . 

PP 

181 

179 

177 

175 

173 

171 

1  68 

166        164 

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16.8 

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50.4 

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72.4 

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67.2 

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86.  S 

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84.0 

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108.6 

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106.3 

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105.0 

103.8 

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100.8 

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126.7 

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119.7      •          .7 

117.6 

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140.0 

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L.  Sin. 

d. 

L.  Tang. 

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8.89464 

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10  402 

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10  080 

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9.99  860 

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9.99859 
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8.91  185 

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8.93  007 

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d. 

L.  Cotg. 

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162 

160 

159 

157 

155 

153 

151 

149 

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120.8 

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L. 

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8.94  195 

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9.99  834 

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144 

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8.95486 

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PP 

145 

143 

141 

139 

138 

136 

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119.7 

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d. 

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8.98  157 

8.98  358 

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114 

0.96  127 

9.99  730 

37 

24 

9  .  o4  7  1  5 

"3 

9.04  987 

0.95  oi3 

9.99728 

36 

25 

9.04828 

9.o5  101 

0.94  899 

9.99  727 

36 

26 

9.04  940 

9.0 

5  214 

"3 

0.94  786 

9.99726 

34 

27 

9-o5  o52 

112 

9.o5  328 

114 

0.94  672 

9.99  724 

33 

28 

g.oS  1  64 

9.o5  44i 

" 

3 

0.94  559 

9.99  723 

32 

29 

9.o5  275 

9.  o5553 

112 

0.94  447 

9.99  721 

3i 

30 

9-o5  386 

9-o5  666 

"3 

0.94334 

9.99  720 

30 

L. 

Cos. 

d. 

L.  Cotg. 

d. 

L. 

Tang. 

L.  Sin. 

83?°  30  . 

PP 

121 

120 

"9 

118 

117 

116 

"5 

114 

"3 

.1 

12.  1 

I2.O 

11.9 

.1 

ii.  8 

ii.7 

ii.  6 

.1 

"•5 

11.4 

"•3 

.2 

24.2 

24.0 

23-8 

2 

23.6 

23-4 

23.2 

.2 

23.0 

22.8 

22.6 

•3 

36.3 

36.0 

35-7 

•3 

35-4 

35-i 

34-8 

•3 

34-5 

34-2 

33-9 

•4 

48.4 

48.0 

47-6 

•4 

47-2 

46.8 

46.4 

•4 

46.0 

4S.6 

45-2 

•  5 

60-5 

60.0 

59-5 

•5 

59-o 

S8.S 

58.0 

•5 

57-5 

56-5 

.6 

72.6 

72.0 

71.4 

.6 

70.8 

70.2 

69.6 

.6 

69.0 

68.4 

67.8 

•7 

84.7 

84.0 

83-3 

•7 

82.6 

81.9 

81.2 

•7 

80.  S 

79.8 

79.1 

8 

96.8 

96.0 

95-2 

.8 

94-4 

93-6 

92.8 

.8 

92.0 

91.2 

90.4 

.9     108.9 

I08.0 

107.1                 .9 

106.2 

105.1 

104.4 

9     I03-5 

IO2.6 

101.7 

6°  3O . 


; 

L. 

Sin. 

d. 

LvTang. 

d. 

L. 

Cotg. 

L. 

Cos. 

30 

9.o5  3«6 

9.06  666 

0.94  334 

9-99  720 

30 

3i 

32 

9-o5  497 
9«o5  607 

IIO 

9-o5  778 
9.o5  890 

112 
112 

0.94  222 

0.94  no 

9.99  718 
9.99  717 

29 

28 

33 

9.o57i7 

no 

9.06  002 

112 

0.93  998 

9.99  716 

27 

34 
35 

g.oS  827 
9.o5937 

IIO 

9.06  1  13 
9.06  224 

III 

o'. 

93887 
33  776 

9.99714 
9.99  713 

26 

25 

36 

9.06  o46 

109 

9.06  335 

III 

o.93  665 

9.99  711 

24 

37 
38 

9.06  i55 
9.06  264 

109 
109 

9.o6445 
9.o6556 

IIO 
III 

0.93555 
0.93444 

9.99  710 
9.99  708 

23 
22 

39 

9.06  372 

108 

9.06  666 

IIO 

o.93  334 

9.99707 

21 

40 

9.06  48i 

109 

9.06  775 

109 

0.93  225 

9.99  705 

20 

4i 

42 

9.06  589 
9.06  696 

107 

9.o6  885 
9.06  994 

109 

0.93  n5 
0.93  006 

9.99  704 
9.99  702 

18 

43 

9.06  8o4 

107 

9.07  io3 

109 

108 

0.92  897 

9.99  701 

17 

44 
45 
46 

9.06  911 
9.07  018 
9.07  124 

107 
106 
107 

9.07  211 

9.07  32O 

9.07  428 

109 
108 
108 

0.92  789 
0.92  680 
0.92  572 

9.99699 
9.99698 
9.99696 

16 
i5 

i4 

47 

48 

9.07  23i 
9.07337 

106 

9.07  536 
9.07  643 

107 

0.92  464 
0.92  357 

9.99695 
9.99  693 

i3 

12 

49 

9.07  442 

105 

infi 

9.07  75i 

0.92  249 

9.99  692 

I  I 

50 

9.07  548 

9.07  858 

107 

0.92  142 

9.99690 

10 

5i 

9.07  653 

105 

9.07  964 

0.92  o36 

9.99689 

9 

62 

9.07  768 

9.08  071 

ir/i 

0.91  929 

9.99687 

8 

53 

9.07863 

105 
105 

9.08  177 

106 

0.91  823 

9.99  686 

7 

54 

9.07  968 

9.08  283 

106 

0.91  717 

9.99  684 

6 

56 

9.08  072 

9.08  389 

0.91  611 

9.99  683 

5 

56 

9.08  176 

9.08495 

0.91  5o5 

9.99  681 

4 

104 

10 

i 

57 

9.08  280 

9.08  600 

0.91  4oo 

9.99  680 

3 

58 

9.08  383 

9.08  705 

3 

0.91  295 

9.99678 

2 

59 

9.08  486 

103 

9.08  810 

105 

0.91  190 

9.99677 

I 

60 

9.08  589 

9.08  914 

0.91  086 

9.99  676 

0 

L. 

Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L 

Sin. 

83°. 

PP 

112 

in 

IIO 

109 

108 

107 

1  06 

105 

104 

.1 

II.  2 

ii.i 

II.  0 

.1 

10.9 

10.8 

10-7 

:i 

10.6 

10.5 

10.4 

.2 

22-4 

22.2 

22.0 

.2 

21.8 

21.6 

21-4 

.2 

21.2 

21.0 

20.8 

3 

33-6 

33-3 

33-o 

•3 

32.7 

32.4 

32.1 

3 

31.8 

31-5 

31.2 

•4 

44.8 

44-4 

44.0 

•4 

43.6 

43-2 

42.8 

•4 

42.4 

42.0 

41.6 

:J 

56.0 
67.2 

55-5 
66.6 

55-° 
66.0 

:I 

54-5 

54-o 
64.8 

£5 

5 
.6 

53-° 
63.6 

52-5 
63.0 

52.0 
62.4 

•7 

78.4 

77-7 

77.0 

•7 

76-3 

75-6 

74-9 

•7 

74-2 

73-5 

72.8 

8 

89.6 

88.8 

88.0 

.8 

87.2 

86.4 

85.6 

.8 

84.8 

84.0 

83-2 

• 

Q 

98.1 

97.2 

96-3 

•9       95-4 

94-5 

93-6 

43 


1 

L. 

Sin. 

d. 

L.  Tang. 

d. 

L. 

Cotg. 

L. 

Cos. 

0 

9.08  589 

103 
103 

IO2 
102 
102 
101 
IO2 
1OI 
101 
1OO 

101 
IOO 
100 

•99 

IOO 

.  99 
99 
98 

99 
98 

98 
98 
98 
97 
97 
97 
97 
96 

97 
96 

9.08  914 

i°s 
104 
104 
103 
104 
103 
103 

102 
103 
102 

102 
101 
102 
IOI 
IOI 
IOI 
IOI 
IOO 
IOO 
IOO 

IOO 

99 
99 
99 
99 
99 
98 

98 
98 
98 

0.91  086 

9.99675 

60 

I 

2 

3 

4 
5 
6 

8 
9 

9.08  692 
9.08  795 
9.08  897 

9.08  999 
9.09  101 
9.09  202 

9.09  3o4 
9.09405 
9.09  5o6 

9.09  019 
9.09  123 
9.09  227 

9.09  33o 
9.09434 
9.o9537 

9.09  64o 
9.09  742 
9.09  84s 

0.90  981 
0.90  877 
0.90  773 

0.90  670 
0.90  566 
0.90  463 

0.90  36o 
0.90  258 
0.90  i55 

9.99  674 
9.99  672 
9.99670 

9.99  669 
9.99  667 
9.99  666 

9.99  664 
9.99  663 
9.99  661 

59 
58 
57 
56 
55 
54 
53 

5l 

10 

9.09  606 

9.09947 

0.90  o53 

9.99659 

50 

1  1 

12 

i3 

i4 
i5 
16 

«7 

18 

J9 

9.09  707 
9,09  807 
9.09907 

9.  TO  OO6 
9.IO  1  06 

9.  10  205 
9.10  3o4 

9-10  4O2 

9.10  5oi 

9.10  049 
9.10  i5o 

9.  IO  252 

9.10353 

9.10  454 

9.10555 

9.10  656 
9.  10  756 
9.10  856 

0.89  g5i 
0.89  850 
0.89  748 

0.89  647 
0.89  546 
0.89445 

0.89  344 
0.89  244 
0.89  i44 

9.99  658 
9.99  656 
9.99655 

9.99  653 
9.99  65i 
9.99  650 

9.99  648 
9.99647 
9-99  645 

49 
48 

47 
46 
45 
44 

43 

42 

4i 

20 

9.10  599 

9.10  g56 

0.89  o44 

9.99  643 

40 

21 
22 
23 

24 
25 
26 

27 
28 
29 

9.10  697 
9.  10  795 
9.  10  893 

9.  10  990 
9.11  087 
9.11  i  84 

9.11  281 

9.11  377 

9.11  474 

9.11  o56 
9.  ii  i55 
9.11  254 

9.1     353 
9.       452 
9.      55i 

9.      649 

9-      ?4? 
9.      845 

0.88  944 
0.88  845 
0.88  746 

0.88  647 
0.88  548 
0.88  449 

0.88  35i 
0.88  253 
0.88  165 

9.99  642 
9.99  64o 
9.99  638 

9.99  637 
9  .99  635 
9.99  633 

9.99  632 
9.99  63o 
9.99  629 

39 

38' 

37 

36 

35 
34 

33 

32 

3i 

30 

9.11  570 

9.1 

I  943 

0.88  057 

9.99627 

30 

L. 

Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L. 

Sin. 

' 

82°  3O  . 

PP 

105 

104 

103 

102 

IOI 

IOO 

99 

98          97 

.1 

.2 

3 

•4 
•  5 
.6 

:i 

i&i 

10.5 

21.0 

31-5 

42.0 

52.5 
63.0 

73-5 

84.0 

04.  5 

10.4 

20.8 

31.2 

41.6 
52.0 
62.4 

72.8 
83.2 

10.3 

20.6 

30-9 
41.2 

I,':i 

72.1 

82.4 

^2^ 

.0 

.0 

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.0 

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•3 
•4 

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9-9 
19.8 
29.7 

39-  6 
49-5 
59-4 

69-3 
79.2 

SQ.I 

9.8         9.7 
19.6       19.4 
29.4       29.1 

39.2       38-8 
49.0       48.5 
58.8       58.2 

68.6       67.9 
78.4       77-6 
88.2        87.3 

.2 

•3 

4 
•5 
.6 

:1 

•9 

20.4 
30.6 

40.8 
51.0 
6l.2 

71.4 
8l.6 

QT.8 

20.2 
3°-3 

40.4 

50-5 
60.6 

70.7 
80.8 

2C 

3C 

4c 

5c 
6c 

70 

8c 

DC 

7°  30 . 


: 

L. 

Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L, 

Cos. 

30 

9.1 

i  570 

9.11  943 

0.88  067 

9-99  627 

30 

3i 

9.11  666 

90 

95 

9.  12  C 

>4o 

97 
08 

0.87  960 

9.99  625 

29 

32 

33 

9.11  761 
9.11  857 

96 

9.12  i  38 

9.  12  235 

97 

0.87  862 
0.87  765 

9.99  624 
9.99  622 

28 
27 

34 

9.11  952 

95 
95 

9.12  332 

97 
06 

0.87  668 

9.99  620 

26 

35 

9.12  047 

9.  12  428 

0.87  572 

9.99  618 

25 

3b 

9.12  142 

95 
94 

9.  12  525 

97 
96 

0.87  475 

9.99617 

24 

38 

9.12  236 
9.12  33i 

95 

9.12  621 
9.12  717 

96 

o.  87379 
0.87  283 

9.99  6i5 
9.99  6i3 

23 
22 

39 

9.12  425 

94 

9.12  8i3 

96 

0.87  187 

9.99  612 

21 

40 

9.12  519 

94 

9.  12  909 

96 

0.87  091 

9.99  610 

20 

4i 

9.12  612 

93 
94 

9  .  1  3  oo4 

95 
qe 

0.86  996 

9.99  608 

'9 

42 

9.12  706 

9.  1  3  099 

0.86  901 

9.99607 

18 

43 

9.12  799 

93 

9  .  1  3  1  94 

95 

0.86  806 

9.99605 

17 

93 

95 

44 

9.12  892 

9-i3  289 

0.86  711 

9.99  6o3 

16 

45 

9.  12  985 

9.  1  3  384 

0.86  616 

9.99  601 

i5 

46 

9-i3  078 

93 

9.13478 

94 

0.86  522 

9.99  600 

i4 

93 

95 

47 
48 

9.i3  171 
9-i3  263 

92 

9.i3573, 
9.13667 

94 

0.86  427 
0.86  333 

9.99  598 
9.99  596 

i3 

12 

49 

9.i3  355 

92 

9.  1  3  761" 

94 

0.86  239 

9.99595 

I  I 

50 

9-i3  447 

92 

9.i3  854 

0.86  i46 

9.99  593 

10 

5i 

9.i3  539 

92 

9.i3  948 

94 
93 

o.86oi 

>2 

9.9959i 

9 

52 

9.  1  3  63o 

9.  1  4  o4i 

o.85  959 

9.99  589 

8 

53 

9.  i3  722 

92 

9.i4  i 

34 

93 
93 

o.85  866 

9.99  588 

7 

54 

9.i38i3 

9.  i4  227 

93 

o.85773 

9.99  586 

6 

55 

9.  1  3  904 

9.i43 

20 

o.85  680 

9.99  584 

5 

56 

9-i3  994 

9° 

9.14^ 

12 

92 

o.85  588 

9.99  582 

4 

91 

92 

57 

9.i4  o85 

9.  i4  5o4 

o.85  496 

9.99  58i 

3 

58 

9.14  175 

9-i4597 

o.854o3 

9-99  579 

2 

59 

9.  i4  266 

91 

9.i4  688 

91 

o.85  3i2 

9.99577 

I 

60 

9.14  356 

9.  i4  780 

o.85  220 

9.99  575 

0 

L. 

Cos. 

d. 

L.  Cotg.       d. 

L.  Tang. 

L. 

Sin. 

V 

82°. 

PP 

97 

96 

95 

94 

93 

92 

91 

90 

.j 

9-7 

9-6 

9-5 

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9-4 

9-3 

9.2 

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9-i 

9.0 

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19.4 

19.2 

19.0 

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18.8 

18.6 

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18.2 

18.0 

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29.1 

28.8 

28.5 

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28.2 

27.9 

27.6 

•3 

27.3 

27.0 

4 

,,8.8 

38.4 

38.0 

•4 

37-6 

37-2 

36.8 

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36-4 

36.0 

5 

48-  s 

48.0 

47-5 

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47.0 

46-5 

46.0 

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45-5 

45-o 

6 

58.2 

57-6 

57-o 

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56.4 

55-8 

55-2 

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54-6 

54-o 

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67.9 
77.6 

67.2 
76.8 

66.5 
76.0 

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65-8 
75-2 

65.1 
74-4 

64.4 
73-6 

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63.7 
72.8 

63.0 
72.0 

9 

87.s 

86.4 

85-5 

•9 

84.6 

83-7 

82.8 

•9 

81.9 

81.0 

45 


, 

L.  Sin. 

d. 

L.  Tang.      d. 

L.  Cotg. 

L.  Cos. 

0 

9.i4356 

0_ 

9.14  780 

o.85  220 

9 

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60 

I 

9-14445 

09 

9° 

9.14  872 

92 

o.85  128 

9 

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59 

2 

9.14535 

9.14  963 

o.85  037 

9 

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58 

3 

9.14624 

89 

9.  1  5  o54 

91 

o.84  946 

9 

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57 

4 

9.14  71 

4 

90 
80 

9.i5  i45 

9i 

o.84  85s 

9 

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56 

5 

9.i48o3 

9.i5  236 

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o.84  764 

9 

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55 

6 

9.14  891 

9.i5  327 

91 

0.84673 

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54 

7 

9.  i4  980 

89 

80 

9.i5  417 

90 

0.84583 

9 

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53 

8 

9.  1  5  069 

9-i5  5o8 

91 

o.84  492 

9 

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52 

9 

9oi5  157 

9..i5  598 

90 

o.84  i 

{02 

9 

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5i 

10 

9.  1  5  245 

88 

9.i5688 

90 

o.84  3i2 

9 

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50 

ii 

9.i5333 

88 

9.i5  777 

59 

0.84  223 

9 

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49 

12 

9.  1  5  421 

9.i5  867 

o.84  i33 

9 

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48 

i3 

9.i5  5o8 

87 

9.  i5  956 

89 

o  .  84  o44 

9 

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47 

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9.  i5  596 

88 
87 

9.1 

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90 
80 

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9 

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46 

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9.i5  683 

Q_ 

9.16  135 

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365 

9 

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45 

16 

9.i5  770 

87 

9.  16  224 

89 
88 

o.83  776 

9 

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44 

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9.i5  857 

87 

9.  16  3i2 

80 

0.83688 

9 

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43 

18 

9.  1  5  944 

86 

9.16  4oi 

o.83  599 

9 

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42 

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9.  16  o3o 

86 

9.  16  489 

o.83 

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9 

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4i 

20 

9.16  1  16 

87 

9.16  577 

88 

00 

o.83  423 

9 

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40 

21 

9.  16  2o3 

86 

9.16  665 

88 

o.83  335 

9 

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39 

22 

9.16  289 

g. 

9.16  753 

o.83  247 

9 

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38 

23 

9.16  374 

86 

9.16  84i 

87 

o.83  159 

9 

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37 

24 

9.16  46o 

j5 

9.  16  928 

88 

o.83  072 

9 

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36 

25 

9.i6545 

86 

9.  17  016 

0.82  984 

9 

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35 

26 

9.16  63i 

8s 

9.  17  io3 

87 
87 

0.82  897 

9 

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34 

27 

9.16  716 

9.17  190 

87 

0.82  810 

9 

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33 

28 

9.16  801 

J 

9.17  277 

0.82  723 

9 

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32 

29 

9.16  886 

85 

9.17  363 

86 

0.82  637 

9 

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3i 

30 

9.  16  970 

84 

9.17450 

87 

0.82  55o 

9 

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30 

L.  Cos. 

d. 

L. 

Cotg. 

d. 

L.  Tang. 

L.  Sin. 

' 

81°3O. 

PP          92 

9» 

9° 

89 

88 

87 

86 

.1         9.2 

9.1 

9.0 

.1 

8.9 

8.8 

.1 

8.7 

8.6 

.2              18.4 

1  8.  2 

18.0 

.2 

17.8 

17.6 

.2 

17.4 

17.2 

•3          27.6 

27-3 

27.0 

•3 

26.7 

26.4 

•3 

26.1 

25.8 

•4          36.8 

36.4 

36.0 

•4 

35-6 

35-2 

•4 

34-8 

34-4 

•  5          46-0 

45-5 

45-o 

.5 

44-5 

44.0 

43-5 

43-° 

•6          55-2 

54-  6 

54-o 

.6 

53-4 

52-8 

.6 

52.2 

51.6 

•7          64-4 

63-7 

63.0 

•7 

62.3 

61.6 

•  7 

60.9 

60.2 

.8          73.6 

72.8 

72.0 

.8 

71.2 

70.4 

.8 

69.6 

68.8 

.9          82.8 

81.9 

81.0 

^—  _ 

78-3 

77-4 

46 


8°  3D . 


; 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

30 

9 

.  16  970 

9.17  450 

0.82  55o 

9 

>99  52O 

30 

3i 

9 

.17055 

85 
8-1 

9.17  536 

86 
86 

0.82  464 

9 

99  5i8 

29 

32 

9 

.17  i39 

9.17  622 

0.82  378 

9 

99  5i7 

28 

33 

9 

.  17  223 

84 
84 

9.17  708 

86 
86 

0.82  292 

9 

99515 

27 

34 

9 

.I73o7 

0  . 

9.17  794 

Rfi 

0.82  206 

9 

995i3 

26 

35 

9 

.i739i 

9.17  880 

O.82   120 

9 

99  5i  i 

25 

36 

9 

.17474 

83 
84 

9.17  965 

85 
86 

0.82  c 

35 

9 

995o9 

24 

37 

9 

.17  558 

9.18  o5i 

0.8  1  949 

9 

99  507 

23 

38 

9.17  64i 

°3 

9.18  i36 

°5 

0.81  864 

9 

.99  5o5 

22 

39 

9 

.17  724 

9.18  221 

85 

0.81  779 

9 

.99  5o3 

21 

40 

9.17807 

83 

9.18  3o6 

°5 

0.8  1  694 

9 

.99  5oi 

20 

4i 

9 

.17  890 

83 
8? 

9.18  391 

85 

0.81  609 

9 

99499 

19 

42 

9 

.17973 

9.18475 

0.81  525 

9 

99497 

1  8 

43 

9 

.i8o55 

82 

9.18  56o 

85 
84 

0.8  1  44o 

9 

.99495 

17 

44 

9 

.18  i37 

0, 

9.18  644 

o  . 

0.81  356 

9 

99494 

16 

45 

9 

.  l8  220 

°3 

9.18  728 

0.81  272 

9 

.99492 

i5 

46 

9.18  3o2 

81 

9.18  812 

84 
84 

0.81  1  88 

9 

.99  490 

i4 

47 

9.18  383 

9.18  896 

o.. 

0.81  io4 

9 

.99488 

i3 

48 

9.18465 

9.18  979 

0.81  021 

9 

.99486 

12 

49 

9.18  547 

82 

9.  19  o63 

84 

0_ 

0.80  937 

9 

•  99484 

I  I 

50 

9.18  628 

9.19  i  46 

°3 
Ro 

0.80  854 

9 

.99  482 

10 

5i 

9.18  709 

81 

9.19  229 

°3 
83 

0.80  771 

9 

.99  48o 

9 

52 

9.  18  790 

9.  19  3i2 

0.80688 

9 

.99478 

8 

53 

9.18  871 

8r 

9.i9395 

83 
83 

0.80  605 

9 

.99476 

7 

54 

9.  18  952 

81 

9.19478 

83 

O.8o  522 

9 

.99474 

6 

55 

9.  19  o33 

9.  19  56i 

P_ 

0.80439 

9 

.99472 

5 

56 

9.19  1  13 

80 

9.19  643 

82 

0.80  357 

9 

.99470 

4 

57 

9.  19  193 

80 

9.19  725 

82 

O.8o  275 

9 

.99  468 

3 

58 

9.19  273 

9.19  807 

O.8o  ] 

93 

9.99  466 

2 

59 

9.19353 

80 

9.  19  889 

82 

0.8o  1 

1  1 

9 

.99  464 

I 

60 

9.19433 

9.19  971 

o.  80  029 

9 

.99  462 

0 

" 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L.  Sin. 

81°. 

PP 

86 

85 

84 

83 

82 

81 

80 

8.6 

8-5 

8.4 

.1 

8.3 

8.2 

>z 

8.1 

8.0 

.2 

17.2 

17.0 

16.8 

.2 

16.6 

,6.4 

.2 

16.2 

16.0 

•3 

25-8 

25-5 

25.2 

•3 

24.9 

24.6 

•3 

24-3 

24.0 

•4 

34-4 

34-° 

33-6 

•4 

33-2 

32.8 

•4 

32-4 

32.0 

•  5 

43-  o 

42-5 

42.0 

•  5 

41-  s 

41.0 

5 

40.5 

40.0 

.6 

51-6 

51-0 

50.4 

.6 

49.8 

49.2 

.6 

48.6 

48.0 

.7 

60.2 

59-5 

58.8 

.7 

58.1 

57-4 

.7 

56-7 

56.0 

.8 

68.8 

68.0 

67.2 

.8 

66.4 

65.6 

.8 

64.8 

64.0 

•9 

7*-4 

•  9          74-7           73-8 

9 

72.9 

72.0 

47 


9°. 


I 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

0 

9.19  433 

9.  19  971 

0.80  029 

9 

.99  462 

60 

I 

9,19  5i3 

9.20  o53 

82 

81 

0.79  947 

9 

.99  46o 

59 

2 

9.19  592 

9.20  1  34 

0.79  866 

9 

.99458 

58 

3 

9.  19  672 

79 

9.20  216 

81 

0.79  784 

9 

.99  456 

57 

4 

9.19  761 

9.20  297 

81 

0.79  703 

9 

•  99  454 

56 

b 

9.19  83o 

9.20  378 

0.79  622 

9 

.99  452 

55 

6 

9.19909 

79 

9.20  459 

0.79  54i 

9 

.99450 

54 

79 

81 

7 

9.1998 

8 

9.20  54o 

0.79  46o 

9 

•  99448 

53 

8 

9.20  067 

9.20  621 

0.79  379 

9 

.99  446 

52 

9 

9.20  i45 

7° 

_Q 

9.20  701 

0 

°-79  299 

9 

•  99  444 

5i 

10 

9.20  223 

9.20  782 

0.79  218 

9 

.99  442 

50 

1  1 

9.20  3o2 

79 
78 

9.20  862 

80 

0.79  i38 

9 

.99  44o 

49 

12 

9.20  38o 

9.20  942 

o.  79  o58 

9 

.99  438 

48 

i3 

9.20458 

7° 
77 

9.21  022 

80 

0.78  978 

9 

.99  436 

47 

i4 

9.20  535 

78 

9.21    102 

80 

0.78  898 

9 

.99434 

46 

i5 

9.20  6i3 

9.21    l82 

0.78  818 

9 

.99432 

45 

16 

9.20  691 

JO 

77 

9.21   26l 

79 
80 

0.78739 

9 

.99  429 

44 

17 

9.20  768 

9.21  34i 

0.78  659 

9 

.99427 

43 

18 

9.20  845 

9.21  420 

0.78  58o 

9 

.99  425 

42 

'9 

9.20  922 

77 

9.21  499 

79 

0.78  5oi 

9 

.99423 

4i 

20 

9.20999 

9.21  578 

79 

o.  78  422 

9 

.99  421 

40 

21 

9.21  076 

77 

9.21  657 

79 

o.78'343 

9 

.99419 

39 

22 

9.21  i53 

•76 

9.21  736 

_0 

0.78  264 

9 

.99417 

38 

23 

9.21  229 

77 

9.21  8i4 

7° 
79 

0.78  186 

9 

.99415 

37 

24 

9.21  3o6 

76 

9.21  893 

78 

0.78  107 

9 

.99  4i3 

36 

25 

9.21  382 

9.21  971 

0.78  029 

9 

.99411 

35 

26 

9.21  458 

76 

9  .  22  o4g 

7° 
78 

o.7795i 

9 

.99  409 

34 

27 

9.21  534 

76 

9-22   127 

78 

0.77873 

9 

.99407 

33 

28 

9,21  610 

9-22  2O5 

_Q 

°-77  795 

9 

.99  4o4 

32 

29 

9.21  685 

•76 

9.22  283 

78 

0.77  717 

9 

.99  402 

3i 

30 

9.21  76 

i 

9.22  36i 

0.77  639 

9 

.  99  4oo 

30 

L.  Cos. 

d. 

L.  Cotg.       d. 

L.  Tang. 

L.  Sin. 

8O°  3O  . 

PP       82 

81 

so 

79 

78 

77 

76 

.1                  8.2 

8.1 

8.0 

.1 

7-9 

7.8 

.  i 

7-7 

7.6 

.2               16.4 

16.2 

16.0 

.2 

15-8 

15.6 

.2 

15-4 

15.2 

3          24.6 

24-3 

24-0 

•3 

23-7 

23-4 

•3 

23.1 

22.8 

•4          32-8 

32.4 

32.0 

•4 

31-6 

31.2 

•4 

30.8 

30-4 

•  5          4i-o 

4°-5 

40.0 

•5 

39-5 

39-  ° 

•5 

38.5 

38.0 

.6          49.2 

48.6 

48.0 

.6 

47-4 

46.8 

.6 

46.2 

45-6 

-7          57-4 

56.7 

56.0 

•7 

55-3 

54-6 

•  7 

53-9 

S3-2 

.8          65.6 

64.8 

64.0 

.8 

63-2 

62.4 

.8 

61.6 

60.8 

•9           73-8 

72.9 

72.0 

•9           ?i-i 

70.2 

•9 

69-3 

68.4 

48 


9°  SO. 


, 

L.  Sin. 

d. 

L.  Tang.      d. 

L.  Cotg. 

L.  Cos. 

30 

9.21  761 

9.22  36i 

0.77  639 

9 

.99  4oo 

30 

3i 

9.21  836 

75 
76 

9.22  438 

77 
78 

0.77  562 

9 

.99398 

29 

32 

9.21  912 

9.22  5i6 

o.77484 

9 

.99  396 

28 

33 

9.21  987 

75 

9.22  5g3 

77 

0.77  407 

9 

•  99  394 

27 

34 

9.22  062 

75 

9.22  670 

77 

0.77  33o 

9 

.99392 

26 

35 

9.22  187 

9.22  747 

0.77  253 

9 

.99  390 

25 

3b 

9.22  211 

74 

9.22  824 

77 

0.77  176 

9 

.99  388 

24 

75 

77 

3? 

9.22  286 

9.22  901 

0.77099 

9 

.99  385 

23 

38 

9.22  36i 

9.22  977 

7 

0.77  O23 

9 

.99  383 

22 

39 

9.22435 

74 

9.23  o54 

77 

0.76  946 

9 

.99  38i 

21 

40 

9.22  5og 

9.23  i3o 

0.76  870 

9 

•99  379 

20 

4i 

9.22  583 

74 
74 

9.23  206 

70 

0.76  794 

9 

•99  377 

19 

42 

9.22  657 

9.23  283 

0.76  717 

9 

.99375 

18 

43 

9.22  731 

74 
74 

9.23  SSg 

7° 
76 

0.76641 

9 

•99  372 

17 

44 

9.22  805 

9.23435 

0.76  565 

9 

.99  370 

16 

45 

9.22  878 

9.23  5io 

0.76  490 

9 

.99  368 

i5 

46 

9.22  952 

74 

9.23  586 

7° 

0.76^ 

n4 

9 

.99  366 

i4 

47 

9.23  O25 

73 

9.23  661 

75 
76 

0.76  339 

9 

.99  364 

i3 

48 

9.23  098 

9.23  737 

0.76  263 

9 

.99  362 

12 

49 

9.23  171 

73 

9.23  812 

75 

0.76  188 

9 

.99359 

I  I 

50 

9.23  244 

9.23  887 

0.76  1  13 

9 

.99  357 

10 

5i 

9.33317 

73 

9.23  962 

75 

0.76  o38 

9 

.99  355 

9 

52 

9.23  390 

9.  24  037 

0.75  963 

9 

.99353 

8 

53 

9.23462 

72 
73 

9.24  112 

74 

0.75888 

9 

.9935i 

7 

54 

9.23535 

9.24  186 

0.75  8i4 

9 

.99348 

6 

55 

9.23  607 

9.24  261 

0.75  739 

9 

.99  346 

5 

56 

9.23  679 

72 

9.24335 

74 

0.75  665 

9 

.99  344 

4 

73 

75 

57 

9.23  752 

9.24  4io 

0.75  590 

9 

.99  342 

3 

58 

9.23  823 

9.24484 

0.75  5i6 

9 

.  99  34x 

» 

2 

59 

9.23  895 

72 

9.24558 

74 

0.75  442 

9 

.99337 

I 

60 

9.23  967 

9.24  632 

o.75368 

9 

.99335 

0 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L.  Sin. 

' 

80°. 

PP          77 

76 

75 

74 

73 

72 

71 

.1         7.7 

•2               15-4 

7.6 
15-2 

7-5 
15-0 

.1 

.2 

a 

,5:1 

.1 

.2 

7-2 

14.4 

14.2 

•3          23.1 

22.8 

22-5 

•3 

22.2 

21.9 

•3 

21.6 

21.3 

•  4          30-8 

3°-4 

30.0 

•4 

29.6 

29.2 

•4 

28.8 

28.4 

•5          38-5 

38-0 

37-5 

•  5 

37-° 

36.5 

•5 

36.0 

35-5 

.6          46.2 

45-6 

45-o 

.6 

44-4 

43-8 

.6 

43-2 

42.6 

•7          53-9 
.8          61.6 

g; 

60.0 

•  7 
.8 

51-8 
59-2 

51-! 

58.4 

:i 

50.4 
57-6 

8:1 

•9           69-3 

68.4 

67-5 

.9          66.6 

65-7 

•9 

64.8 

63.9 

I 

L.  Sin. 

d. 

L. 

Tang.     d. 

L.  Cotg. 

L.  Cos. 

d. 

0 

9.23  967 

9- 

24  632 

74 
73 
74 
73 
74 
73 
73 
73 
73 
73 
72 
73 
72 
73 
72 
72 
72 
72 
72 
71 
72 
71 
72 

71 
7° 

0.75  368 

9-99 

335 

60 

I 

2 

3 

4 
5 
6 

7 
8 

9 

9.24  no 
9.24  181 

9.24253 
9.24  324 
9.24395 

9.24466 
9.24536 
9.24  607 

71 

71 
72 

71 

71 
70 

71 

9- 
9- 
9- 

9- 
9- 
9- 

9- 
9- 
9- 

24  706 

24  779 
24853 

24  026 
25  ooo 
25  073 

25^219 
2*5  292- 

0.75  294 

O.75  221 

0.75  147 

0.75  074 
0.75  ooo 

0.74927 

o.  74854 
0.74  781 
0.74  708 

ooo  ooo  ooo 

ooo  ooo  ooo 
ooo  ooo  ooo 

333 
33i 
328 

326 

324 

322 

3i9 
3i7 
3i5 

2 

3 

2 
2 
2 

3 

2 
2 
2 

3 

2 
2 
2 

3 
2 

2 

3 

2 
2 

2 

3 

2 
2 

3 

2 
2 

3 

2 
2 

59 

58 
5? 

56 

55 
54 
53 

52 

5i 

10 

9.24677 

7° 

9- 

25365 

o.74635 

9.99 

3i3 

50 

II 

12 

i3 

U 

i5 
16 

17 

18 

9.24  748 
9.24818 
9.24888 

9.24  g58 
9.25  028 
9.25  098 

9.25  168 
9.25  237 
9.25  307 

70 
70 
70 
70 
70 
70 
69 
70 

000  000  000 

25437 
25  5io 
25  582 

25655 
25  727 
25  799 

25  871 

25  943 

26  015 

0.74563 
0.74  490 
0.74418 

0.74345 
0.74273 

0.74  201 
O.74  129 

0.74  057 
0.73  985 

ooo  ooo  ooo 

ooo  ooo  ooo 
ooo  ooo  ooo 

3io 
3o8 
3o6 

3o4 
3oi 
299 

297 
294 
292 

49 

48 

47 
46 
45 
44 

43 

42 

4i 

20 

9.25  376 

09 

9- 

26  086 

0.73  914 

9.99 

290 

40 

21 
22 
23 

24 
25 

26 

27 
28 
29 

9.25445 
9.25  5i4 
9.25  583 

9-25652 
9.25  721 
9.25  790 

9.25  858 
9.25927 
9.25  995 

°9 

68 
69 
68 
68 

ooo  ooo  ooo 

26  i58 
26  229 
26  3oi 

26  372 
26443 
265i4 

26  585 
26655 
26  726 

0.73  842 
o.7377i 
0.73  699 

0.73  628 
o.73557 
0.73  486 

o!73345 
o.73  274 

9.99 
9.99 
9.99 

9-99 
9.99 

9.99 

9.99 
9.99 
9-99 

288 
285 
283 

281 
278 
276 

274 
271 
269 

39 
38 
37 

36 
35 

34 
33 

32 

3i 

30 

9.  26  o63 

9- 

26  797 

0.73  2o3 

9.99 

267 

30 

L.  Cos. 

d. 

L. 

Cotg. 

d. 

L.  Tang. 

L.  Sin. 

d. 

79°  30'. 

PP 

.2 

•3 
•4 

i 

.9 

74 

73            72 

.1 

.2 

•3 
•4 

71 

70 

69 

.2 

•3 

•  4 
•  5 
.6 

68              3 

22.2 
29.6 

44-4 

7-3            7-2 
14.6          14.4 
21.9         21.6 

29.2         28.8 
36.5          36.0 
43-8          43-2 

Si.  i          5°-4 
58.4          57.6 
65.7          64.8 

14.2 
21.3 

28.4 
35-5 
42.6 

7.0 

21.  0 
28.0 

35-o 
42.0 

49-o 
56.0 

6.9 
13-8 
20.7 

27.6 
34-5 
41.4 

48-3 
SS-2 
62.1 

6.8          0.3 
13.6           0.6 
20.4           0.9 

27.2                1.2 

34.0                1.5 

40.8           1.8 

47.6                 2.1 

54-4            2.4 
61.2            2.7 

5o 


3O. 


/ 

L.  Sin. 

d. 

L.  Tang. 

d. 

L. 

Cotg. 

L.  Cos. 

d. 

30 

9.26  o63 

9.26 

797 

o. 

73  2o3 

9-99 

267 

30 

3i 

9.26  i3i 

68 

9.26 

867 

70 

o. 

73  i33 

9.99 

264 

2 

29 

32 

9.26  199 

9.26  937 

0. 

73o63 

9.99 

262 

28 

33 

9.26  267 

9.27 

008 

71 

o. 

72  992 

9.99 

260 

27 

34 

9-26335 

68 
68 

9.27 

078 

7o 

0.72  922 

9.99 

257 

3 

2 

26 

36 

9.26  4o3 

9.27 

i48 

7° 

0.72  852 

9.99 

255 

25 

36 

9.26  470 

67 

9.27 

218 

70 

0.72  782 

9.99 

252 

24 

37 

9.26  538 

68 

67 

9.27 

288 

70 
60 

0.72  712 

9.99 

25o 

2 

23 

38 

9.26  6o5 

9.27 

357 

o. 

72643 

9  .  99  248 

22 

39 

9.26  672 

07 
fit 

9.27  427 

70 

o. 

72573 

9.99 

245 

21 

40 

9 

.26  739 

07 

9.27  496 

69 

0. 

72  5o4 

9.99 

243 

20 

4i 

9 

.26  806 

6? 

9.27 

566 

7° 
60 

o. 

72434 

9.99 

241 

3 

19 

42 

9 

.26873 

9.27 

635 

0.72  365 

9.99 

238 

1  8 

43 

9.26  940 

07 

67 

9.27 

704 

69 
69 

0. 

72  296 

9-99 

236 

3 

17 

44 

9 

.27007 

66 

9.27 

773 

60 

o. 

72  227 

9.99 

233 

2 

16 

45 

9.27073 

9.27 

842 

0. 

72  i58 

9.99 

23l 

i5 

46 

9.27  i4o 

67 
66 

9.27 

911 

69 
69 

o. 

72  089 

9.99 

229 

i4 

47 

9.27  206 

67 

9.27 

980 

60 

0. 

72  020 

9.99 

226 

2 

i3 

48 

9.27  273 

9.28 

049 

o. 

71  95i 

9.99 

224 

12 

49 

9.27  339 

9.28 

117 

o. 

71  883 

9.99 

221 

I  I 

50 

9 

.27  405 

9.28 

186 

69 

0. 

71  8i4 

9.99 

219 

10 

5i 

9.2747i 

66 

9.28 

254 

60 

0. 

71  746 

9.99 

2I7 

3 

9 

52 

9.2-7  537 

9.28 

323 

o. 

71  677 

9.99 

2l4 

8 

53 

9.27  602 

65 

9.28 

39i 

o. 

71  6o9 

9.99 

212 

7 

66 

68 

3 

54 

9.27  668 

66 

9.28459 

68 

o. 

71  54i 

9.99209 

2 

6 

55 

9.27734 

9.28 

527 

0. 

7i473 

9.99 

207 

5 

56 

9.27799 

65 

9.28 

595 

0. 

71  4o5 

9.99 

204 

4 

65 

67 

57 

9.27  864 

66 

9.28 

662 

68 

o. 

71  338 

9.99 

202 

2 

3 

58 

9  .  27  g3o 

9.28 

73o 

0. 

71  2-70 

9.99 

2OO 

2 

59 

9.27  995 

9.28  798 

67 

o. 

7i  202 

9.99 

197 

I 

60 

9.28  060 

°S 

9.28  865 

o. 

71  135 

9.99 

195 

0 

L.  Cos.    |  d. 

L.  Cotg.      d. 

L. 

Tang. 

L.  Sin.     d. 

' 

79°. 

PP 

70 

69 

68 

67 

66 

65              3 

.1 

7.0 

6.9 

6.8 

.1         6.7 

6.6 

.1 

6.5          0.3 

.2 

14.0 

13-8 

13-6 

.2               13.4 

13.2 

.2 

13.0           0.6 

•3 

21.0 

20.7 

20.4 

,  3          20.  i 

19.8 

•3 

19.5           0.9 

•4 

28.0 

27.6 

27.2 

.4      26.8 

26.4 

•4 

26.O               1.2 

•5 

35-° 

34-5 

34-o 

•5           33-5 

33-o 

•5 

32-5                1-5 

.6 

42.0 

41.4 

40.8 

.6          40.2 

39-  6 

.6 

39.0           1.8 

•  7 

49.0 

48-3 

47.6 

•7           46-9 

46.2 

•  7 

45-5           2.1 

.8 

56.0 

55-2 

54-4 

•8           53-6 

52.8 

.8 

52.0           2.4 

•9 

63.0 

62.1 

•9           6o-3 

59-4 

58-5           2.7 

5i 


11°. 


' 

L.  Sin. 

d. 

L. 

Tang.     d. 

L.  Cotg. 

L.  Cos.    i  d. 

0 

9.28  060 

65 
65 
64 
65 
65 
64 
64 

65 
64 

9 

28  865 

68 
67 
67 
67 
67 
67 
67 
67 
66 
67 

66 
67 
66 
66 
66 
66 
66 
66 
66 
65 
66 
65 
65 
66 

65 
65 
65 
65 
64 

0.71  135 

9.99  195 

3 

2 

3 

2 

3 

2 

3 

2 

3 

2 

3 

2 

3 

2 

3 

2 

3 

2 

3 

60 

J 

2 

3 

4 
5 
6 

8 
9 

9.28  125 
9  .28  190 
9.28  254 

9.28  319 
9.28  384 
9.28448 

9.28  5i2 
9.28577 
9.28  64  1 

9 
9 
9 

9 
9 
9 

9- 
9- 
9- 

28  933 
29  ooo 
29  067 

29  1  34 

29  201 
29  268 

29  335 
29  4O2 

29  468 

o.  71  067 
0.71  ooo 
0.70  933 

0.70866 
0.70  799 
0.70  732 

0.70  665 
0.70  598 
0.70  532 

OOO  OOO  OOO 

ooo  ooo  ooo 

192 
I9O 

I87 

185 
182 

180 
177 

172 

59 
58 
5? 
56 
55 
54 
53 

52 

Si 

10 

9.28  705 

9.29535 

0.70  465 

9.99 

170 

50 

1  1 

12 

i3 

i4 
i5 
16 

18 
'9 

9.28  769 
9.28  833 
9.28  896 

9.28  960 
9.29  024 
9.29  087 

9.29  i5o 
9.29  214 
9.29  277 

04 
64 

63 
64 
64 
63 
63 
64 

63 

9 
9- 
9- 

9- 
9- 
9- 

9- 
9- 
9 

29  601 
29  668 
29  734 

29  800 
29  866 
29932 

29998 
3oo64 
3o  i3o 

0.70  399 
0.70  332 
0.70  266 

0.70  200 
0.70  1  34 
0.70068 

0.70  002 
0.69  936 
0.69  870 

9.99 
9.99 
9.99 

9.99 
9.99 
9-99 

9-99 
9-99 
9.99 

167 
165 
162 

1  60 
157 

162 
150 

i47 

49 

48 

4? 
46 
45 
44 

43 

42 

4i 

20 

9.29  34o 

03 

9- 

3o  i95 

0.69  805 

9.99 

145 

3 

2 

3 

2 

3 

2 

3 
3 

2 

40 

21 
22 
23 

24 
25 

26 

27 
28 
29 

9.29  4o3 
9.29  466 
9.29  529 

9.29  591 
9  .29  654 
9.29  716 

9-29779 
9.29  84i 
9.29  903 

63 
63 
63 
62 

63 
62 
63 
62 
62 

9 
9- 
9- 

9- 
9- 
9- 

9- 
9- 
9- 

3o  261 
3o326 
3o  391 

30  522 

3o587 

3o652 
3o7i7 
3o  782 

0.69  739 
0.69  674 
0.69  609 

o,69  543 
0.69  478 
0.69  4i3 

0.69  348 
o.69  283 
0.69  218 

ooo  ooo  ooo 

ooo  ooo  ooo 
ooo  ooo  ooo 

142 
i4o 
i37 

132 

i3o 

127 
124 

122 

39 

38 

37 
36 
35 
34 

33 

32 

3i 

30 

9.29  966 

63 

9- 

3o846 

0.69  1  54 

9.99 

119 

30 

L.  Cos. 

d. 

L 

,Cotg 

. 

d. 

L.  Tang. 

L.  Sin. 

d. 

' 

78°  30  . 

PP 

t.         .2 

3 
•4 

.u 

68 

67          66 

.2 

•3 

•4 
•  5 
.6 

.7 
.8 

65 

64 

63 

62             3 

6.8 
13-6 
20.4 

27.2 
34-o 
40.8 

47.6 
54-4 

6.7         6.6 
13.4       13.2 
20.  i       19.8 

26.8       26.4 
33-5       33-o 
40.2       39.6 

46.9       46.2 
53-6       52-8 
60.3       59.4 

6-5 
13-0 

19-5 

26.0 
32-5 

45-5 
52.0 

58.  s 

6.4 

12.8 

19.2 

256 

32.0 

38.4 
44.8 

63 

12.6 

18.9 

25.2 

37-8 

44.1 
50-4 

.2 

•3 

•4 

mti 

6.2            .3 
12.4            .6 
18.6            .9 

24.8          1.2 
31.0          1.5 
37.2          1.8 

43-4           2.1 
49.6         .2.4 
5S.8            2.7 

1 1°  3D . 


L.  Sin.       d. 

L. 

Tang. 

d. 

L.  Cotg. 

L.  Cos.     d. 

30 

9.29  966 

9 

3o  846 

0.69  1  54 

9.99  119 

30 

3i 

32 

9-3o  028 
9»3o  090 

62 

9 
9- 

3o  91  1 
3o  975 

64 

0.69  089 
0.69  025 

9.99 
9-99 

117 

u4 

3 

29 

28 

33 

9.3o 

i5i 

9- 

3  1  o4o 

65 

0.68  960 

9.99 

I  12 

2 

27 

34 

9.3o  2i3 

62 
62 

9- 

3i  io4 

0.68  896 

9.99 

109 

3 

26 

35 

9.30275 

9- 

3i  168 

0.68  832 

9.99 

1O6 

3 

25 

36 

9.3o 

336 

9- 

3i  233 

65 

0.68  767 

9.99 

104 

2 

24 

37 

9-3o  398 

62 
61 

9- 

3i  297 

64 

0.68  703 

9.99 

1OI 

3 

23 

38 

9«3o  459 

9- 

3i  36i 

4 

0.68  639 

9.99 

099 

22 

39 

9  .  3o 

521 

61 

9- 

3i  425 

64 

0.68  575 

9-99 

096 

3 

21 

40 

9.3o  582 

61 

9- 

3i  489 

04 

6? 

0.68  5n 

9.99 

093 

3 

20 

4i 

9.3o643 

61 

9- 

3i  552 

61 

0.68448 

9.99 

091 

I9 

42 

9.3o  704 

61 

9- 

3i  616 

fio 

0.68  384 

9-99 

088 

18 

43 

9.  3o  765 

61 

9- 

3i  679 

°3 

0.68  32i 

9.99 

086 

3 

17 

44 

9-3o  826 

61 

9- 

3i743 

6-j 

0.68  257 

9.99 

o83 

16 

45 

9.30887 

9- 

3  1  806 

0.68  194 

9.99 

080 

i5 

46 

9.30947 

61 

9- 

3i  870 

64 

0.68  i3o 

9.99 

078 

3 

i4 

47 

9.  3  1  008 

60 

9- 

3i  933 

0.68  067 

9.99 

076 

i3 

48 

9.3i  068 

9- 

3  1  996 

°3 

0-68  oo4 

9.99072 

12 

49 

9.3i 

129 

60 

9- 

32  o5c 

1 

°3 

0.67  941 

9.99070 

1  I 

50 

9.  3i 

189 

61 

9- 

32   122 

fio 

0.67  878 

9-99 

067 

3 

10 

5i 

9.3i  250 

60 

9- 

32  i85 

°3 
63 

0.67815 

9.99  o64 

9 

52 

9-3i  3io 

60 

9- 

32  248 

0.67  752 

9-99 

062 

8 

53 

9.3i  370 

60 

9- 

32  3u 

63 
62 

0.67  689 

9.99 

059 

3 

7 

54 

9.  3  1  43o 

60 

9- 

32373 

0.67  627 

9.99 

066 

6 

55 

9.  3  1  490 

9- 

32436 

3 

0.67  564 

9.99 

o54 

5 

56 

9.3i  549 

59 
60 

9- 

32498 

62 

0.67  5o2 

9.99 

o5i 

3 

4 

57 

9.  3  1  609 

60 

9- 

32  56i 

62 

0.67  439 

9.99 

o48 

2 

3 

58 

9.  3  1  669 

9 

32  623 

o.67377 

9.99 

o46 

2 

59 

9-3i 

728 

59 

9- 

32685 

62 

o.673i5 

9.99 

o43 

3 

I 

60 

9.3i 

788 

9 

32747 

o.67253 

9,99 

o4o 

0 

L.  Cos.    !  d. 

L, 

Cotg 

.     d. 

L.  Tang. 

L.  Sin. 

d. 

' 

78°. 

PP 

65 

64         63 

6a 

61 

60 

59           3 

.1 

6.5 

6.4          6.3 

.! 

6.2 

6.1 

6.0 

.  i 

5-9         0.3 

.2 

13.0 

12.8             12.6 

.2 

12.4 

12.2 

I2.O 

.2 

ii.  8         0.6 

•3 

19.2      18.9 

•  3 

18.6 

l8-3 

18.0 

•3 

17.7         0.9 

4 

26.0 

25.6       25.2 

-4 

24.8 

24.4 

24.0 

4 

23.6              1.2 

32.5 

32.0      31-5 

.5 

31.0 

3°-5 

30.0 

.5 

29-5          1-5 

.6 

38.4         37-8 

.6 

37-2 

36.6 

36.0 

.6 

35-4         1-8 

.7 

45-5 

44.8        44.1 

•7 

43-4 

42.7 

42.0 

•7 

41-3              2.1 

.8 

52.0 

51.2         50.4 

.8 

49-6 

48.8 

48.0 

.8 

47.2        2.4 

•9        58-5 

54-9 

54-°                -9 

S3-1           2.7 

53 


12C 


, 

L.  Sin. 

d. 

L. 

Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

0 

9.3i  788 

9- 

32  747 

61 

0.67  253 

9.99040 

60 

I 

9.3i  847 

60 

9- 

32  810 

62 

0.67  190 

9.99 

o38 

3 

59 

2 

9.3i  907 

9- 

32  872 

0.67  128 

9.99 

o35 

58 

3 

9-3i  966 

59 

9- 

32  933 

62 

0.67  067 

9.99 

032 

3 

2 

$7 

4 

9.32  025 

59 

9- 

32  995 

62 

0.67  005 

9.99 

o3o 

56 

5 

9.32  o84 

9- 

33o57 

0.66  943 

9.99 

027 

55 

6 

9.32  i43 

59 

9- 

33  119 

61 

0.66  881 

9.99 

024 

3 

2 

54 

7 

9.32  202 

59 

9- 

33  180 

62 

0.6 

6  820 

9.99 

022 

53 

8 

g.32  261 

_0 

9- 

33242 

0.66  758 

9.99 

019 

52 

9 

9.32  319 

9- 

33  3o3 

0.66  697 

9.99 

016 

3 

5i 

10 

9.32  378 

59 

9- 

33  365 

0.66635 

9-99 

oi3 

50 

ii 

9.32437 

58 

9- 

33426 

61 

0.66  574 

9.99 

on 

49 

12 

9.32  L 

fo5 

9- 

33487 

o.665i3 

9.99 

008 

48 

i3 

9.32  553 

58 

9- 

33548 

61 

0.66  452 

9.99 

oo5 

3 

47 

59 

61 

3 

i4 

9.32  612 

9- 

33609 

61 

o.6639i 

9.99 

002 

46 

i5 

9.32  670 

9- 

33  670 

0.66  33o 

9.99 

ooo 

45 

16 

9.32  728 

58 

9- 

3373i 

61 

0.66  269 

9.98 

997 

3 

44 

58 

61 

3 

17 

9.32  786 

9- 

33  792 

61 

0.66  208 

9.98 

994 

3 

43 

18 

9.32  844 

9- 

33853 

0.66  i47 

9.98 

991 

42 

r9 

9.32  902 

S8 

9- 

33  9i3 

60 

0.66  087 

9.98 

989 

4i 

20 

9.32  960 

S8 

9- 

33974 

61 

0.66  026 

9.98 

986 

40 

21 

9.33  018 

58 

9- 

34o34 

61 

o.65  966 

9.98 

983 

3 

39 

22 

9.33  075 

9 

34  095 

o.65  905 

9.98 

98o 

38 

23 

9.33  i33 

58 

9- 

34  1  55 

o.65  845 

9.98 

978 

37 

57 

60 

3 

24 

9-33  190 

eg 

9- 

342i5 

61 

o.65  785 

9.98 

975 

36 

25 

9.33  248 

9- 

34276 

o.65  724 

9.98 

972 

35 

26 

9.33  3o5 

57 

9- 

34336 

o.65  664 

9.98 

969 

3 

34 

57 

60 

2 

27 

9.33  362 

,0 

9- 

34396 

60 

o.65  6o4 

9.98 

967 

33 

28 

9.33  420 

9- 

34456 

0.65544 

9.98 

964 

32 

29 

9-33477 

57 

9.345i6 

0.65484 

9.98 

96i 

3 

3i 

30 

9.33  534 

57 

9- 

34576 

o.65  424 

9.98 

958 

30 

L.  Cos. 

d. 

L. 

Cotg. 

d. 

L.  Tang. 

L.  Sin.     d. 

77°  3O. 

PP 

63 

62          61 

60 

59 

58 

57             3 

., 

6.3 

6.2          6.1 

.! 

6.0 

5-9 

5-8 

.! 

5-7           0.3 

.2 

12.6 

12-4            12.2 

2 

12.0 

ii.  8 

n.6 

2 

11.4          0.6 

•3 

18.9 

18.6        18.3 

•  3 

18.0 

17.7 

17.4 

3 

17.1           0.9 

•4 

25.2 

24.8        24.4 

4 

24.0 

23.6 

23.2 

4 

22.8                 1.2 

•5 

31.5 

31.0        30.5 

5 

30.0 

29-5 

29.0 

5 

28.5           1.5 

.6 

37-8 

37.2        36.6 

6 

36.0 

35-4 

34-  8 

6 

34.2           1.8 

.7 

44.1 

43-4        42-7 

7 

42.0 

4i-3 

40.6 

•7 

39.9                2.  I 

.8 

50.4 

49.6        48.8 

8 

48.0 

47.2 

46.4 

.8 

45-6           2.4 

9        56-7 

55-8         54-9 

9 

54-° 

53-  ! 

52.2 

.9 

5r-3            2.7 

54 


12°  30 '. 


; 

L.  Sin. 

d. 

L.  Tang. 

d. 

L 

,  Cotg. 

L.  Cos. 

d. 

30 

9.33  534 

57 

9.34 

576 

o.65  424 

9.98  958 

30 

3i 

9.33  591 

S6 

9-34 

635 

60 

o. 

65  365 

9.98955 

3 

29 

32 

9.33647 

9-34 

695 

60 

0. 

653o5 

9.98 

953 

28 

33 

9-33  704 

57 

9-34 

755 

59 

o. 

65245 

9.98 

950 

3 

27 

34 

9.33  761 

57 

9.34 

8i4 

60 

o. 

65  186 

9.98  947 

26 

35 

9.338i8 

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9.34 

874 

0. 

65  126 

9.98 

944 

25 

36 

9.33874 

5° 

9-34 

933 

59 

o. 

65  067 

9.98 

94  1 

3 

24 

37 
38 

9.3393i 
9.33  987 

57 
56 

efi 

9.34 
9.35 

992 
o5i 

59 
59 
60 

o. 
o. 

65  008 
64  949 

9.98  938 
9.98  936 

3 

2 

23 
22 

39 

9.34o43 

5° 

9.35 

in 

0. 

64  889 

9.98 

933 

3 

21- 

40 

9.34  100 

cfi 

9.36 

170 

o. 

6483o 

9.98  930 

20 

4i 

9.34i56 

5° 

9.35 

229 

59 

o. 

64  771 

9.98 

927 

3 

19 

42 

9.34  212 

5 

9.35 

288 

0. 

64  712 

9.98 

924 

18 

43 

9.34268 

56 
56 

9.35 

347 

59 
58 

o. 

64653 

9.98 

921 

3 

2 

i? 

44 
45 

9.34324 
9.34380 

56 

9.35 
9.35 

4o5 

464 

59 

o.64  595 
o.64  536 

9.98 
9.98 

919 
916 

3 

16 
i5 

46 

9.34436 

56 

9-35 

523 

59 

o. 

64477 

9.98  913 

3 

i4 

55 

58 

3 

47 
48 

9.34491 
9.34547 

56 

9.35 
9.35 

58i 
64o 

59 

o. 
o. 

644i9 
64  36o 

9.98  910 
9.98  907 

3 

i3 

12 

49 

9.34  602 

55 

9.35 

698 

58 

o. 

64  3o2 

9.98 

904 

3 

I  I 

50 

9.34658 

5° 

9.35 

757 

59 

eg 

0. 

64243 

9.98 

901 

3 

10 

5i 

9.34  71 

1 

56 

9.35 

815 

58 

o. 

64  1  85 

9.98 

898 

2 

9 

52 

9.34  769 

9.35 

873 

o. 

64  127 

9.98 

896 

8 

53 

9.34  824 

55 
55 

9.35 

58 
58 

o. 

64  069 

9.98 

893 

3 

7 

54 

9.34879 

55 

9.35 

989 

eg 

o. 

64  01  1 

9.98 

890 

3 

6 

55 

9.34934 

9.36  047 

o. 

63953 

9.98 

887 

5 

56 

9.34989 

55 

9.36 

io5 

58 

0. 

63895 

9.98 

884 

4 

57 

9.35  o44 

55 

9.36 

i63 

58 

eg 

0. 

63837 

9.98 

881 

3 
3 

3 

58 

9-35  099 

9.36 

221 

o. 

63779 

9.98 

878 

2 

59 

9.35  1  54 

55 

9.  36 

279 

58 

o. 

63  721 

9.98 

875 

I 

60 

9.35  209 

55 

9.  36 

336 

57 

0. 

63  664 

9.98 

872 

0 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L. 

Tang. 

L.  Sin. 

d. 

' 

77V 

. 

PP       60 

59 

58 

57 

56 

? 

55              3 

.1            6.0 

5-9 

5-8 

•i            5-7 

5-6 

j 

5-5           0.3 

.2               I2.O 

n.8 

11.6 

.2              II-4 

II.  2 

2 

ii.  o           0.6 

.3         18.0 

17.7 

17.4 

•3          i7-i 

16.8 

3 

16.5           0.9 

4         24.0 

23.6 

23-2 

.4               22.8 

22-4 

4 

22.0                1.2 

•5          3°-° 

29-5 

29.0 

•5          28.5 

28.0 

5 

27-5           i-5 

.6         36.0 

35-4 

34-8 

.6         34.2 

33-6 

6 

33-o           1.8 

.7         42-0 

4x-3 

40.6 

•7          39-9 

39-2 

7 

38-5           2.1 

.8         48.0 

47-2 

46.4 

.8          45-6 

44-8 

8 

44.0           2.4 

•9           54-° 

S3-  1            52-2 

•9     1      5i-3 

50.4                  9 

49-5            2-7 

55 


' 

L.  Sin. 

d. 

L.  Tang-. 

d. 

L. 

Cotg. 

L.  Cos.     d. 

0 

9.  35  209 

9.36 

336 

0. 

63  664 

9.98 

872 

60 

I 

9.35  263 

55 

9.  36  394 

5° 
c8 

o. 

63  606 

9.98  869 

59 

2 

9.35  3i8 

9.  36 

452 

0. 

63  548 

9.98 

867 

58 

3 

9.35373 

9.  36 

509 

57 

o. 

63491 

9.98 

864 

3 

57 

54 

57 

- 

4 

9.35427 

54 

9.  36 

566 

rg 

o. 

63434 

9.98 

861 

56 

5 

9-35481 

9.36 

624 

o  . 

63  376 

9.98 

858 

55 

6 

9.35536 

55 

9.  36 

681 

57 

0. 

63  3i9 

9.98 

855 

: 

54 

7 

9.35  590 

54 

9.36 

738 

57 

0. 

63  262 

9.98 

852 

3 

53 

8 

9.35  644 

9-36 

795 

0. 

63  205 

9.98 

849 

52 

9 

9.35  698 

54 

9.36 

852 

57 

0. 

63  i48 

9.98 

846 

3 

5i 

10 

9.35  752 

9.36 

9°9 

57 

o. 

63  091 

9.98 

843 

" 

50 

ii 

9.35  806 

54 

9.36 

966 

57 

0. 

63o34 

9.98 

84o 

3 

49 

12 

9.35  860 

9-37 

023 

o. 

62  977 

9.98 

837 

48 

i3 

9.35  914 

54 

9.37 

080 

57 

o. 

62  920 

9.98 

834 

- 

> 

47 

i4 

9.  35  968 

54 

9.37 

i37 

57 
cfi 

0. 

62863 

9.98 

83i 

3 

46 

i5 

9-36  022 

9.37 

I93 

5° 

0. 

62  8o7 

9.98828 

45 

16 

9.36  075 

53 

9.37 

250 

57 

o. 

62  75o 

9.98825 

• 

i 

44 

17 

9.36  129 

54 

9.37 

3o6 

5t> 

o. 

62  6g4 

9.98 

822 

3 

43 

18 

9.36  182 

9.37 

363 

57 

0. 

62  637 

9.98 

8i9 

42 

19 

9.  36  236 

54 

9.37 

4i9 

56 

o. 

62  58i 

9.98816 

3 

4i 

20 

9.36  289 

53 

9.37 

476 

57 

o. 

62  524 

9.98 

8i3 

• 

> 

40 

21 

9.  36  342 

53 

9.37 

532 

56 

o. 

62468 

9.98 

810 

39 

22 

9.36395 

53 

9.37 

588 

56 

o. 

62  412 

9.98 

8o7 

38 

23 

g.36  449 

54 

9.37 

644 

56 

o. 

62  356 

9.98 

8o4 

3 

37 

24 

9-36  5o2 

53 

9.37 

700 

56 

0. 

62  3oo 

9-98 

801 

3 

36 

25 

9.36555 

53 

9.37 

756 

56 

o. 

62  244 

9.98 

798 

35 

26 

9.36  608 

53 

9.37 

812 

50 

o. 

62  188 

9.98 

795 

• 

34 

27 
28 

~9?3\66o 
9.36  7i3 

52 
53 

9.37 
9.37 

868 
924 

56 
56 

o. 

0. 

62  1  32 
62  o76 

9.98 
9.98 

792 

789 

3 

3 

33 

32 

29 

9.  36  766 

53 

9.37 

980 

56 

0. 

62  020 

9.98 

•786 

3i 

30 

9.36  819 

53 

9.  38 

o35 

55 

0. 

61  965 

9.98 

783 

30 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L. 

Tang. 

L.  Sin. 

d. 

- 

76°  3O. 

PP       58 

57 

56 

55 

54 

53 

3 

5-8 

5-7 

5.6 

•  i            5-5 

5-4 

.1 

5-3 

•  3 

.2              II.  6 

n-4   . 

II.  2 

.2              II.  0 

10.8 

.2 

10.6 

.6 

•3          17-4 

17.1 

1  6.  8 

•3          16.5 

16.2 

•3 

159 

•9 

•4          232 

22.8 

22.4 

.4               22.O 

21.6 

•4 

21.2 

1.2 

•  5          29.0 

28.5 

28.0 

•5          27.5 

27.0 

.5 

26.5 

i-5 

.6          34-8 

34-2 

33-6 

•  6          33.0 

32-4 

.6 

31.8 

1.8 

.7          4°-6 

39-9 

392 

•7           38.5 

37-8 

•7 

37-  * 

2.1 

.8          46.4 

45-6 

44-8 

.8          44.0 

43-2 

.8 

42.4 

2-4 

•9           52-2 

51.3     !     50.4 

•9          49-5 

48.6 

47-7 

2.7 

56 


13°3O 


' 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

30 

9.36 

819 

9 

.38  o35 

0.61  965 

9.98 

783 

30 

3i 

32 

9-36 
9-36 

87I 
924 

52 

53 

9 
9 

.38  091 
.38  147 

56 

0.61  909 
0.61  853 

9.98 
9.98 

780 
777 

3 
3 

29 

28 

33 

9.36 

976 

52 

9 

.38  202 

55 

0.61  798 

9.98 

774 

3 

27 

34 

9.37 

028 

52 

9 

.38257 

55 
eft 

0.61  743 

9.98 

771 

3 

26 

35 

9.37 

08  1 

9 

.383i3 

5° 

0.61  687 

9.98 

768 

3 

25 

36 

9.37 

i33 

52 

9 

.38368 

55 

0.61  632 

9.98 

765 

3 

24 

37 

9.37 

i85 

52 

9 

.38423 

55 

0.61  577 

9.98 

762 

3 

23 

38 

9-37 

237 

9 

.38479 

J 

0.61  52i 

9.98 

759 

3 

22 

39 

9.37 

289 

52 

9 

.38  534 

55 

0.61  466 

9.98 

756 

3 

21 

40 

9.37 

34i 

9 

.38  589 

0.61  4n 

9.98 

753 

3 

20 

4i 

9.37 

393 

52 

9 

38  644 

55 

0.61  356 

9.98 

75o 

3 

'9 

42 

9.37 

445 

9 

38  699 

0.61  3oi 

9.98 

746 

18 

43 

9.37 

497 

9 

38754 

0.61  246 

9.98 

743 

3 

ij 

52 

54 

3 

44 

9.37 

549 

51 

9 

38  808 

0.61  192 

9.98 

74o 

16 

45 

9-37 

600 

9 

38863 

0.61  137 

9.98 

737 

i5 

46 

9.37  652 

9 

38  918 

55 

0.61  082 

9.98 

734 

J 

i4 

47 

9-37 

7o3 

51 
52 

9 

38  97 

2 

54 

0.61  028 

9.98 

73i 

3 

i3 

48 

9.37 

755 

9 

39  027 

0.60  973 

9.98 

728 

12 

49 

9.37  806 

51 

9 

39  082 

55 

0.60  9i8 

9.98 

725 

3 

II 

50 

9.37  858 

9 

39  1  36 

54 

0.60  864 

9.98 

722 

3 

10 

5i 

9.3-7909 

51 

9 

39  190 

0.60  810 

9.98 

719 

9 

52 

9.3-7  960 

9 

39  24 

5 

0.60  755 

9.98 

7i5 

8 

53 

9-38  on 

9 

39299 

54 

0.60  701 

9.98 

712 

3 

7 

51 

54 

3 

54 

9.38  062 

9 

39  353 

0.60  647 

9.98 

7°9 

6 

55 

9.  38 

n3 

9 

39  4o 

7 

0.60  593 

9.98 

7o6 

5 

56 

9.38 

1  64 

51 

9 

3946i 

54 

0.60  539 

9.98 

7o3 

3 

4 

51 

54 

3 

57 

9.38 

2l5 

9 

39  5i5 

0.60485 

9.98 

7oo 

3 

58 

9.38  266 

9 

39  569 

o.6o43i 

9.98 

697 

2 

59 

9.  38 

3i7 

51 

9 

39623 

54 

0.60  377 

9.98 

694 

3 

I 

60 

9.  38 

368 

51 

9 

39677 

54 

0.60  323 

9.98 

69o 

4 

0 

L.  Cos.   d. 

L.  Cotg. 

d. 

L.  Tang. 

L.  Sin. 

d. 

' 

76°. 

PP 

56 

55 

54 

53 

52 

51 

4     3 

! 

5.6 

5-5 

5-4 

.1 

5-3 

5-2 

5.1 

.1 

0.4    0.3 

2 

II.  2 

II.  0 

10.8 

.2 

10.6 

10.4 

IO.2 

.2 

0.8    0.6 

3 

16.8 

16.5 

16.2 

•3 

15-9 

15.6 

15-3 

•3 

1.2       0.9 

4 

22.4 

22.  0 

21.6 

•4 

21.2 

20.8 

20.4 

•4 

1.6       1.2 

5 

28.0 

27.5 

27.0 

•  5 

26.5 

26.0 

25-5 

•5 

2.O       1.5 

6 

33-6 

33-0 

32.4 

.6 

31-8 

31.2 

30.6 

.6 

2.4     1.8 

7 

39-2 

38.  s 

37-8 

•  7 

37-1 

*4 

35-7 

•  7 

2.8       2.1 

8 

44-8 

44.0 

43-2 

.8 

42.4 

41.6 

40.8 

.8 

3.2       2.4 

9    5°-4 

49.5    48.6 

47-7 

46.8 

45-9       -9 

3.6       2.7 

14°. 


f 

L.  Sin. 

d. 

L. 

Tang. 

d. 

L.  Cotg. 

L.  Cos.     d. 

0 

9.  38  368 

9- 

39677 

54 
54 
53 
54 
53 
54 
53 
54 

53 

54 
53 
53 
53 
53 
53 
53 
52 
53 

53 
52 

52 
53 
52 
52 
S2 
53 
52 

0.60  323 

9.98 

690 

60 

2 

3 

4 
5 
6 

7 
8 

9 

9.384i8 
9.38469 
9.  38  5i9 

9.38  570 
9.  38  620 
9.  38  670 

9.38  721 
9.  38  771 
9-38  821 

51 
50 

51 

5° 
5° 
Si 
50 
So 

9- 
9- 
9- 

9- 
9- 
9- 

9- 
9- 
9- 

3973i 
39785 
39838 

39  892 
39945 
39999 

4o  o52 
4o  1  06 
4o  i59 

0.60  269 
0.60  2i5 
0.60  162 

0.60  1  08 
0.60  055 
0.60  ooi 

0.59  948 
0.59  84i 

9.98  687 
9.98  684 
9.98  681 

9.98  678 
9.98675 
9.98  671 

9.98668 
9.98  665 
9.98  662 

3 
3 
3 
3 
4 
3 
3 
3 

59 

58 

56 
55 
54 
53 

52 

5i 

10 

9-38  871 

9- 

4o  212 

0.59  788 

9.98 

659 

50 

1  1 

12 

i3 

i4 
i5 
16 

«7 

18 

'9 

9-38  921 
9.38  971 
9.39  02  1 

9.39  071 

9.39  121 
9.39  170 

g.Sg  22O 
9.39270 

9.393i9 

So 
So 
50 
50 
49 
So 
5° 
49 

9- 
9- 
9- 

9- 
9- 
9- 

9- 
9- 
9- 

4o  266 
4o  3i9 
40372 

40425 
40478 
4o53i 

4o584 
4o636 
40689 

o.59  734 
o.5968i 
0.59  628 

o.Sg  575 

O.Sg  522 
O.Sg  469 

0.59  4i6 
0.59  364 
o.Sg  3i  i 

9.98 
9.98 
9.98 

9.98 
9.98 
9.98 

9.98 
9.98 
9.98 

656 
652 
649 

646 
643 
64o 

636 
633 
63o 

4 
3 
3 
3 
3 
4 
3 

3 

49 

48 

4? 
46 

44 
43 

42 

4i 

20 

9.39  369 

5° 

9- 

4o  742 

0.59  258 

9.98 

627 

40 

21 
22 
23 

24 
25 

26 

27 
28 
29 

9.39  4i8 
9.39467 
9.39  5i7 

9.39  566 
9.39615 
9.39  664 

9.397i3 
9.39  762 
9.39  8n 

49 
49 
5° 
49 
49 
49 
49 
49 
49 
49 

9- 
9- 
9- 

9- 
9- 
9- 

9- 

9- 
9- 

40795 
40847 
4o  900 

4o  952 
4i  005 
4i  057 

4i  109 
4i  161 
4i  214 

o.Sg  2o5 
0.59  i53 
0.59  100 

0.59048 
o.58  995 
0.58943 

o.58  891 
o.58  839 
o.58  786 

9.98 
9.98 
9.98 

9.98 
9.98 
9.98 

9.98 
9.98 
9.98 

623 
620 
617 

6i4 
610 
607 

6o4 
601 

597 

3 
3 
3 
4 
3 
3 
3 
4 

39 
38 
3? 

36 
35 

34 
33 

32 

3i 

30 

9.39  860 

9- 

4i  266 

o.58  734 

9.98 

594 

3 

30 

L.  Cos. 

d. 

L, 

Cotg. 

d. 

L.  Tang. 

L.  Sin. 

d. 

' 

75°  30  . 

PP 

.2 

•3 
•4 

:I 

•7 

-8 

•  q 

54 

53          53 

.1 

.2 

•3 
•4 

:I 
:l 

~ 

51 

50 

49 

4       |      3 

,11 

16.2 

21.6 

27.0 
32-4 

37-8 
|f 

53          5-2 
10.6         10.4 
15.9         15.6 

21.2            20.8 

26.  5        26.0 

31.8       31.2 

37-1         36-4 
42.4        41.6 

47-7         46.8 

IO.2 

15-3 
20.4 

25  I 
30.6 

35-7 
40.8 

45  9 

5-o 

IO.O 

15-0 

20.0 
25.0 
30.0 

35-o 
40.0 

4.9 
9.8 
14.7 

19.6 
24-5 
29-4 

34-3 
39-2 
44.1 

.2 

•3 
•4 

1 

:i 

•  9 

:J       :i 

1.2                       .9 

1.6                   1.2 
2.0                   I.5 

2.4             1.8 

2.8                  2.1 

3.2        2.4 

3.6          2.7 

58 


14°3O 


> 

L.  Sin. 

d. 

L. 

Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

30 

9.39  860 

49 

9.41  266 

o.58  734 

9.98  594 

30 

3i 

9.39909 

49 

9.41  3i8 

52 

o.58  682 

9.98  591 

29 

32 

9.39  958 

48 

9.41  370 

o.5863o 

9.98  588 

28 

33 

9.40  006 

49 

9.41  422 

52 

o.58  578 

9.98  584 

^ 

27 

34 

9.40  c 

55 

48 

9.41  4?4 

s2 

o.58  526 

9.98  58  r 

3 

26 

35 

9.40  io3 

9- 

4i  526 

0.58474 

9.98 

578 

25 

36 

9.40  i 

52 

9.41  578 

0.58422 

9.98574 

24 

48 

5* 

3 

37 

9.40  200 

49 

9- 

4i  629 

s2 

o.58  37i 

9.98 

57i 

3 

23 

38 

9.40  249 

9- 

4r  681 

o.58  319 

9.98  568 

22 

39 

9.40  297 

4° 

9- 

4i  733 

52 

o.58  267 

9.98 

565 

3 

21 

40 

9.40  346 

.0 

9- 

4i  784 

o.58  216 

9.98 

56i 

20 

4i 

9.4o  3 

94 

48 

9- 

4i  836 

o.58  i64 

9.98 

558 

3 

19 

42 

9.40  442 

9- 

4i  887 

o.58  n3 

9.98 

55,5 

18 

43 

9.40490 

48 

9- 

4i  939 

52 
51 

o.58  061 

9.98 

55i 

ll 

44 

9.40  538 

9- 

4i  990 

o.58  oio 

9.98 

548 

3 

16 

45 

9.40  586 

9- 

42  o4i 

0.57  959 

9.98 

545 

i5 

46 

9.40  634 

48 

9- 

42  093 

52 

0.57  907 

9.98 

54i 

i4 

48 

51 

3 

47 

9.40  682 

4.8 

9- 

42  i44 

o.57856 

9.98 

538 

3 

i3 

48 

9.40  730 

9- 

42  195 

0.57  805 

9.98 

535 

12 

49 

9.40778 

48 

9- 

42  246 

51 

0.57  754 

9.98 

53i 

3 

I  I 

50 

9.40  825 

47 

.0 

9- 

42  297 

51 

0.57  703 

9.98 

528 

10 

5i 

9.40  873 

48 

9- 

42  348 

51 

0.57  652 

9.98 

525 

4 

9 

52 

9.40  921 

9- 

42  399 

0.57  601 

9.98 

521 

8 

53 

9.40968 

47 
48 

9- 

42  45o 

Si 
51 

0.57  550 

9.98 

5i8 

3 

7 

54 

9.41  016 

9? 

42  5oi 

0.57  499 

9.98 

515 

4 

6 

55 

9.41  o63 

9- 

42  552 

o.57448 

9.98 

5n 

5 

56 

9.41  in 

4° 

9- 

426o3 

51 

o.57397 

9.98 

5o8 

4 

57 

9.41  i58 

47 

9- 

42653 

So 

o.57347 

9.98 

505 

3 
4 

3 

58 

9.41  2o5 

9- 

42  704 

0.57  296 

9.98 

5oi 

2 

59 

9-4l   252 

47 

9- 

42755 

bl 

0.57  245 

9.98 

498 

I 

60 

9.41  3oo 

40 

9- 

42  8o5 

5° 

o.57  195 

9.98 

494 

0 

" 

L.  Cos. 

d. 

L. 

Cotg. 

d.     L.  Tang. 

L.  Sin. 

d. 

75°. 

PR 

52 

51        50 

49 

48 

47 

4 

3 

.! 

52 

5.1       5.0 

.1         4.9 

4.8 

4-7 

.1 

0.4 

°-3 

2 

10.4 

10.2             10.  0 

.2             9.8 

9.6 

9.4 

.2 

0.8 

0.6 

3 

15-6 

15-3             15° 

3       14-7 

14.4 

14.1 

•3 

1.2 

0.9 

4 

20.8 

20.  4         20.  o 

.4       19.6 

19.2 

18.8 

•4 

1.6 

1.2 

I 

26.0 

31  2 

25.5         25.0 
30.6         30.0 

•  5       24.5 
.6       29.4 

24.0 
28.8 

23-5 
28.2 

•5 
.6 

2.O 
2.4 

1^8 

7 

36.4 

35-7        35-o 

•7       34-3 

33-6 

32-9 

•  7 

2.8 

2.1 

8 

41.6 

40.8        40.0 

.8       39.2 

38-4 

37-6 

.8 

3-2 

2-4 

.9       46.8 

459         45'° 

.9        44.1         43.2 

42.3              .9 

59 


15°. 


• 

L.  Sin.       d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos.      d. 

0 

9-4i 

3oo 

9.42  bo5 

51 
50 
51 
50 
50 
51 
50 
50 
50 
5° 

5° 

5° 
50 
50 
50 
49 
So 
5° 
49 
50 

49 
5o 
49 
5° 
49 
49 
49 
5° 
49 
49 

o.57  195 

9.98  494 

60 

2 

3 

4 
5 
6 

7 

8 

9 

9-4i 
9>4i 
9.41 

9,41 
9.41 
9-4i 

9-4i 
9.41 
9.41 

34? 
394 
44  1 

488 
535 
582 

628 
670 
722 

47 
47 
47 
47 
47 
46 

47 
47 

9.42  856 
9.42  906 
9.42  957 

9.43  007 
9.43o57 
9.43  108 

9.43i58 
9.43  208 
9.43258 

0.57  1  44 
0.57  094 
0.57  043 

0.56993 
o.56  943 
o.56  892 

0.56842 
o.56  792 
o.  56  742 

9.98  491 
9.98  488 
9.98484 
9.98  481 

9.98477 
9.98  474 

9.98  471 
9.98  467 
9.  98  464 

3 
4 
3 
4 
3 
3 
4 
3 

59 

58 
5? 
56 
55 
54 

53 

52 

5i 

10 

9.41 

768 

47 
46 

47 
46 

47 
46 
46 
47 
46 

9.43  3o8 

o.56  692 

9 

.  98  460 

50 

1  1 

12 

i3 

i4 
i5 
16 

17 
18 

19 

9.41 
9.41 

9-4i 

9.41 
9.42 

9.42 

9.42 
9.42 
9.42 

815 
861 
908 

954 

OOI 

o47 

ogS 

i4o 

186 

9.43358 
9.434o8 
9-43458 

9.435o8 
9.43558 
9.43  607 

9.43657 
9.43  707 
9.43756 

0.56642 
o.56  592 
o.,  56  542 

o.56  492 
0.56442 
0.56393 

0.56343 
o.56  293 
o.56  244 

9 
9 
9 

9 
9 
9 

9 
9 
9 

.98457 
.98453 
.98450 

.98447 
.98443 
.98440 

.98436 
.98433 

.98  429 

4 
3 

4 
3 
4 
3 

4 

49 

48 

4? 

46 
45 
44 

43 

42 

4i 

20 

9.42 

232 

46 

9.43  806 

o.56  194 

9 

.98426 

4 
3 
4 
3 
3 
4 
3 
4 
3 
4 

40 

21 
22 
23 

24 
25 

26 

27 
28 
29 

9.42 
9.42 
9.42 

9.42 
9.42 
9.42 

9.42 
9.42 
9.42 

278 
324 
37o 

4i6 
46  1 
507 

553 
599 

644 

46 
46 
46 
46 
45 
46 
46 
46 
45 
46 

9.43855 
9.43  905 
9.43954 

9.44  oo4 
9.44  o53 

9.44  102 

9.44  i5i 
9.44  20  1 
9-44  250 

o.56  i4§ 
o.56  095 
o.56  o46 

o.55  996 
o.55  947 
o.55  898 

0.55849 
o.55  799 
0.55  75o 

9 
9 
9 

9 
9 
9 

9 

9 
9 

.98  422 

.98419 
.98415 

.98412 
.98  409 
.98  4o5 

.98  402 
.98  398 
.98395 

39 
38 
37 
36 
35 
34 
33 

32 

3i 

30 

9.42 

690 

9.44  299 

o.55  701 

9 

.98  391 

30 

L.  Cos. 

d. 

L.  Cotgr.      d. 

L.  Tang. 

L.  Sin. 

d. 

74°  3D  '. 

PP 

.1 

.2 

•3 
•4 

:l 

:i 

51 

50 

49 

48         47         46 

.1 

.2 

•3 

•4 

•  5 
.6 

:l 

•9 

45 

4            3 

5-i 

10.2 
15-3 

20-4 
25.5 
30.6 

35-7 
40.8 

45  9 

5-o 

10.0 

15-0 

20.  o 

25-0 
30.0 

35-o 
40.0 

45.0 

49 
9.8 
14.7 

19.6 
24-5 
29.4 

34-3 
39-2 
44.1 

.2 

-3 

•4 
•5 
.6 

3 

4.8        4-7         4-6 
9.6        9.4         9.2 
14.4       14.1        13.8 

19.2       18.8       18.4 
24.0      23.5       23.0 
28.8       28.2       27.6 

33-6       32.9       32.2 
38.4       37.6       36.8 
43.2       423        41  4 

4-5 
9.0 
i3-5 

18.0 
22.5 
27.0 

3'-5 
36.0 

4°-5 

0.4         0.3 
0.8         0.6 
1.2         0.9 

1.6               1.2 
2.0              1.5 

2.4          1.8 

2.8               2.1 

3.2        2.4 

3.6        2.7 

60 


15°  3O 


L.  Sin.       d. 

L 

.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

30 

9.42 

690 

9 

.44  299 

o.55  701 

9.98 

39i 

30 

3i 

9.42 

735 

46 

9 

.44348 

49 

0.55652 

9.98 

388 

3 

29 

32 

9.42  781 

9 

.44397 

o.556o3 

9.98 

384 

28 

33 

9.42 

826 

46 

9 

.44446 

0.55554 

9.98 

38i 

3 
4 

27 

34 

9.42  872 

45 

9 

.44495 

49 

o.555o5 

9.98 

377 

26 

35 

9.42 

917 

9 

.44544 

Q 

0.55456 

9.98 

373 

25 

36 

9.42 

962 

45 

9 

.44  592 

o.  554o8 

9.98 

370 

3 

24 

46 

49 

37 

9.43  008 

9 

,4464i 

o.55359 

9.98 

366 

23 

38 

9-43o53 

9 

44  690 

.0 

o.55  3io 

9.98 

363 

22 

39 

9-43 

098 

45 

9 

44738 

o.55  262 

9.98 

359 

4 

21 

40 

9.43 

i43 

45 

9 

.44  787 

o.55  2i3 

9.98 

356 

20 

4i 

9-43 

188 

45 

9 

.44836 

o.55  1  64 

9.98 

352 

4 

'9 

42 

9.43 

233 

9 

4488 

4 

1 

o.55  116 

9.98 

349 

18 

43 

9.43 

278 

45 

9 

44933 

49 

o.55  067 

9.98 

345 

4 

17 

44 

9-43 

323 

45 

9 

.44981 

48 

o.55  019 

9.98 

342 

3 

16 

45 

9.43 

367 

9 

45  029 

1 

0.54  971 

9.98 

338 

i5 

46 

9.43412 

45 

9 

45  078 

49 

o.54  922 

9.98 

334 

4 

i4 

47 

9.43457 

45 

9 

.45  126 

4H 

0.54874 

9.98 

33i 

3 

i3 

48 

9.43 

502 

9 

45  174 

0.54826 

9.98 

327 

12 

49 

9-43 

546 

44 

9 

.45  222 

48 

o.54  778 

9.98 

324 

3 

I  I 

50 

9.43 

59i 

45 

9 

.45  271 

49 

o.54  729 

9.98 

320 

10 

Si 

9.43635 

44 

9 

.453l9 

48 

o.5468i 

9.98 

3i7 

3 

9 

52 

9.43  680 

45 

9 

.45367 

4 

0.54633 

9.98 

3i3 

8 

53 

9-43 

724 

44 

.4541 

s 

48 

0.54585 

9.98 

309 

4 

7 

54 

9-43 

769 

45 

9 

.45463 

48 

.0 

0.54537 

9.98 

3o6 

3 

6 

55 

9.43 

8i3 

44 

9 

.455n 

0.54489 

9.98 

302 

5 

56 

9.43 

857 

44 

9 

.45559 

48 

o.5444i 

9.98299 

i 

4 

57 

9.43 

901 

44 

9 

.456o6 

47 

.Q 

o.54394 

9.98  295 

4 

3 

58 

9-43 

946 

45 

9 

.45654 

0.54346 

9.98 

291 

2 

59 

9.43 

990 

44 

9 

.45  702 

o.54  298 

9.98  288 

3 

I 

60 

9-44 

o34 

44 

9 

.45750 

4° 

0.54  25o 

9.98 

284 

0 

L.  Cos. 

d. 

L 

.  Cotg. 

d. 

L.  Tang. 

L.  Sin. 

d. 

' 

74° 

. 

PP 

49 

48            47 

46 

45 

44 

4 

3 

.1 

4-9 

4-8          4-7 

,i 

4.6 

4-5 

4-4 

.1 

0.4 

°'3 

.2 

9.8 

9.6          9.4 

.2 

9-2 

9.0 

8.8 

.2 

0.8 

0.6 

•3 

14.7 

14.4         14.1 

•3 

13-8 

13-5 

13-2 

•3 

1.2 

0.9 

•4 

19.6 

19.2        18.8 

•4 

18.4 

18.0 

17.6 

•4 

1.6 

1.2 

f  c 

24-5 

24.0        23.5 

.  tj 

23.0 

22.5 

22.0 

•  5 

2.0 

1.5 

.6 

29.4 

28.8        28.2 

.6 

27.6 

27.0 

26.4 

.6 

2.4 

1.8 

34-3 

33-6        32-9 

•7 

32.2 

3i-5 

30.8 

•7 

2.8 

2.1 

39  2 

38-4        37-6 

.8 

36.8 

36.0 

35-2 

.8 

3-2 

2.4 

•9        44-  x 

43  .2         4^ 

41.4 

40-5 

39-6                 -9 

3-6 

2-7 

61 


' 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

0 

9.44  o34 

9.45  750 

o.54  25o 

9 

.98  284 

60 

I 

9.44078 

9.45  797 

47 

0.54  203 

9 

.98  281 

59 

2 

9.44  122 

9.45845 

4 

o.54i55 

9 

.98277 

58 

3 

9.44  1  66 

44 

9.45  892 

47 

o.54  108 

9 

.98273 

57 

44 

48 

3 

4 

9  .  44  2  1  o 

9.45  940 

o.54  060 

9 

.98  270 

56 

5 

9.44253 

9.45987 

o.54oi3 

9 

.98  266 

55 

6 

9.44  297 

44 

9.46  035 

48 

o.53965 

9 

.98262 

54 

7 

9.4434i 

44 

9.46  082 

47 
j.8 

o.53  918 

9 

.98  259 

3 

53 

8 

9.44385 

9.46  i3o 

o.53  870 

9 

.98  255 

52 

9 

9.44428 

43 

9-46  177 

47 

0.53823 

9 

.98  261 

5i 

10 

9.44  472 

9.46  224 

47 

o.53  776 

9 

.98  248 

50 

ii 

9.44  5i6 

9.46  271 

47 

.0 

o.53  729 

9 

.98244 

49 

12 

9.44559 

9.46  319 

0.5368T 

9 

.98  240 

48 

i3 

9-44  602 

43 

9.46  366 

47 

0.53634 

9 

.98237 

3 

47 

i4 

9-44646 

44 

9-464i3 

47 

o.53587 

9 

.98  233 

4 

46 

i5 

9.44689 

9-46  46o 

47 

o.53  54o 

9 

.98  229 

45 

16 

9.44733 

44 

9-46  507 

47 

0.53493 

9 

.98  226 

3 

44 

17 

9-44  776 

43 

9.46554 

47 

0.53446 

9 

.98  222 

4 
4 

43 

18 

9.44  819 

9.46  601 

47 

o.53  399 

9 

.98  218 

42 

'9 

9.44  862 

43 

9.46648 

47 

o.53  352 

9 

.98215 

3 

4i 

20 

9.44  go5 

9.46  694 

46 

o.533o6 

9 

.98  211 

4 

40 

21 

9.44948 

9.46  741 

47 

o.53  259 

9 

.98  207 

3 

39 

22 

9.44  992 

9.46  788 

47 

O.53  212 

9 

.98  204 

38 

23 

9.45  035 

43 

9.46835 

47 

o.53i65 

9 

.98  2OO 

4 

37 

24 

9.45077 

42 

9.46  881 

46 

o.53  119 

9 

.98   196 

4 
4 

36 

25 

9.45    120 

43 

9.46  928 

47 

o.53  072 

9.98  192 

35 

26 

9.45  i63 

43 

9.46975 

47 

o.53  025 

9 

.98   189 

34 

27 

9.45  206 

43 

9-47  021 

46 

o.52  979 

9 

.98   185 

4 

33 

28 

9.45  249 

9.47  068 

47 

o.52  932 

9 

.98  181 

32 

29 

9.45  292 

43 

9-47 

i4 

46 

o.52  886 

9 

.98  177 

3i 

30 

9.45334 

42 

9.47  160 

46 

o.52  84o 

9 

.98  174 

30 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L.Sin.    Id. 

' 

73°  3D  . 

PP 

48 

47 

46 

45 

44 

43 

42 

4            3 

,! 

4.8 

4-7, 

4.6 

.1 

4-5 

4-4 

4-3 

.i 

4.2 

0.4         0.3 

.2 

q.6 

9-4\ 

9.2 

.2 

q.o 

8.8 

8.6 

.2 

8.4 

0.8          0.6 

•3 

14.4 

14.1 

13-8 

•3 

13-5 

13.2 

12.9 

•3 

12.6 

1.2          0.9 

•4 

19.2 

18.8 

18.4 

•4 

18.0 

17.6 

17.2 

•4 

16.8 

1.6               1.2 

.6 

24.0 
28.8 

S5 

23-0 
27.6 

:! 

22.5 
27.0 

22.0 
26.4 

21-5 

25-8 

•5 
.6 

21.  0 
25.2 

2.0              J-5 

2.4          1.8 

•7 

336 

32.9 

32.2 

.7 

§*.5 

30.8 

30.1 

•  7 

29.4 

2.  8               2.  I 

.8 

38.4 

37-6 

36.8 

.8 

36.0 

35-2 

•tf-4 

.8 

33-6 

3.2               2.4 

•9       43.2 

42.3       41-4 

40.  s        3Q.6         38.7 

37.8 

3.6               2.7 

16°  30 '. 


L.  Sin. 

d. 

L.  Tang.     d. 

L.  Cotg. 

L.  Cos. 

d. 

30 

9.45  344 

9.47  160 

o.52  84o 

9 

.98  174 

30 

3i 

32 

9.45  419 

43 
42 

9.47  207 
9.47  253 

47 
46 

o.52  793 
o.52  747 

9 
9 

.98  170 
.98  1  66 

4 
4 

29 

28 

33 

9-45 

462 

43 

9.47299 

46 

o.52  701 

9 

.98  162 

4 

27 

34 
35 

9-45 
9.45 

5o4 
547 

42 
43 

9.47346 
9-47392 

47 
46 

o.52  654 

0.52  608 

9 
9 

.98  159 
.98  i5§ 

3 
4 

26 

25 

36 

9.45 

589 

9-47438 

46 

o.52  562 

9 

.98  i 

5i 

4 

24 

37 

9-45 

632 

43 

9-47484 

46 
46 

o.52  5i6 

9 

.98147 

4 

23 

38 

9-45  674 

9.47  53o 

o.52  470 

9 

.98  1  44 

3 

22 

39 

9.45 

716 

42 

9.47  576 

46 

O.52  424 

9 

.98  i4o 

4 

21 

40 

9.45 

758 

9.47  622 

4° 

o.52378 

9 

.98  1  36 

4 

20 

4i 

9-45 

801 

42 

9.47668 

46 

o.52  332 

9 

.98  i32 

4 

19 

42 

9-45 

843 

9-47  7*4 

0.52  286 

9 

.98  129 

18 

43 

9-45 

885 

42 

9.47  760 

46 

0.52  24O 

9 

.98125 

4 

17 

44 

9.45 

927 

42 

9.47  806 

.f. 

o.52  194 

9 

.98  121 

16 

45 

9-45 

969 

9.47852 

0.52  li 

18 

9 

.98   117 

i5 

46 

9-46 

OI  I 

9.47897 

45 

o.52  io3 

9 

.98  ii3 

4 

i4 

47 

9.46o53 

42 

9-47943 

46 
.f. 

O.52  of 

>7 

9 

.98  no 

3 

i3 

48 

9.46  095 

9.47989 

4° 

O.52  Oil 

9 

.98  106 

12 

49 

9-46 

i36 

41 

9.48  035 

4'' 

o.5i  965 

9 

.98  102 

4 

I  I 

50 

9-46 

178 

9.48  080 

45 

o.5i  920 

9 

.98  098 

4 

10 

5i 

9-46 

220 

42 

9.48  126 

o.5i  874 

9 

.98  094 

4 

9 

52 

9-46 

262 

9.48  171 

o.5i  829 

9 

.98  090 

8 

53 

9-46 

3o3 

9.48  217 

40 

o.5i  783 

9 

.98  087 

3 

7 

54 

9-46 

34.5 

42 

9.48  262 

45 

o.5i  738 

9 

.98083 

4 

6 

55 

9.46  386 

9.48  307 

o.5i  693 

9 

.98  079 

5 

56 

9-46 

428 

42 

9.48  353 

46 

o.5i  647 

9 

.98  075 

4 

4 

4> 

45 

4 

57 

9.46469 

9.48  398 

o.D  i  602 

9 

.98  071 

3 

58 

9-46 

5ii 

9-48443 

o.5i  557 

9 

.98  067 

2 

59 

9.46 

552 

41 

9.48489 

46 

o.5i  5ii 

9 

.98  o63 

4 

I 

60 

9-46 

594 

9.48  534 

45 

o.5i  466 

9 

.98  060 

0 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L,  Tang. 

L.  Sin. 

d. 

' 

73°. 

PP 

47 

46 

45 

44 

43 

42 

41 

4            3 

.2 

4-7 
9-4 

4-6 
9.2 

4-5 
9.0 

.2 

a 

tl 

t: 

.1 

2 

4.1 

8.2 

0.4         0.3 
0.8         0.6 

3 

14.1 

i3-8 

13-5 

•3 

13-2 

12.9 

12.6 

•3 

12.3 

1.2              0.9 

•4 

18.8 

18.4 

18.0 

•4 

17.6 

17.2 

16.8 

•4 

16.4 

1.6              1.2 

•5 
.6 

23-5 
28.2 

23.0 
27.6 

•22.5 
27.0 

.6 

22.O 
26.4 

21.5 
25.8 

21.0 
25.2 

•5 
.6 

20.5 

24.6 

2.0              1.5 

2.4         1.8 

•  7 

32.9 

32.2 

3i  5 

•7 

30.8 

30.1 

29.4 

•7 

28.7 

2.8              2.1 

.8 

37-6 

36.8 

36.0 

.8 

35-2 

34-4 

33-6 

.8 

32.8 

3.2      2.4 

•9       42.3 

41.4       40.5 

39-6       38.7        37.8 

63 


17 


f 

L.  Sin. 

d. 

L. 

Tang 

. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

0 

9  .46  594 

9- 

48  534 

45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
44 
45 
45 
44 
45 
44 
45 
44 
45 

44 
45 
44 
44 
45 
44 
44 
44 
44 
44 

o.5i  466 

9  .  98  060 

60 

I 

2 

3 

4 
5 
6 

8 
9 

9-46635 
9.46  676 
9.46717 

9.46  758 
g.46  800 
9.46  84i 

9.46  882 
9.46  923 
9.46  964 

41 
41 

42 
41 
4r 

40 
41 
41 
41 

40 
40 

40 

41 

40 

4* 

40 

4° 
41 

40 
40 
40 

9.48  579 
-£•48  624 
9.48  669 

9.48  714 
9-48  759 
9.48  8o4 

9.48849 
9.48  894 
9.48  939 

o.5i  421 
o.5i  376 
o.5i  33i 

o.5i  286 
o.5i  241 
o.5i  196 

o.5i  i5i 
o.5  1  1  06 
o.5i  06  1 

9.98  o56 
9.98o52 
9.98  o48 

9.98  o44 
9.98  o4o 
9.98  o36 

9.98  o32 
9.98  029 
9.98  025 

4 
4 
4 
4 
4 
4 
3 
4 

59 

58 
57 

56 
55 
54 
53 

52 

5i 

10 

9.47  005 

9- 

48  984 

o.5i  016 

9.98 

O2I 

50 

1  1 

12 

i3 

i5 
16 

17 
18 

'9 

9.47  o45 
9.47  086 
9.47  127 

9.47  168 
9.47  209 
9.47  249 

9.47  290 
9.47371 

9.49  029 
9.49  073 
9.49  118 

9.49  i63 
9.49  207 

9.49  252 

9.49  296 
9.49  34i 
9.49  385 

o.5o  971 
o.5o  927 
o.5o882 

o.5o  837 
o.5o  793 

0.50748 

o.5o  704 
o.5o  65g 
o.5o  615 

9.98  017 
9.98  oi3 
9.98  009 

9.98  oo5 
9.98  ooi 
9.97  997 

9.97993 
9.97989 
9.97986 

4 
4 
4 
4 
4 
4 
4 
3 
4 

4 
4 
4 
4 
4 
4 
4 
4 
4 
4 

49 
48 

47 
46 
45 
44 

43 

42 

4i 

20 

9.47  4i  i 

9- 

49  43o 

o.5o  570 

9-97 

982 

40 

21 
22 
23 

24 
25 

26 

27 
28 
29 

9.47  452 
9.47  492 
9.47533 

9.47  6i3 
9.47654 

9.47  694 
9.47  734 
9.47  774 

9.49  474 
9  .49  5ig 
9.49  563 

9.49  607 
9.49  652 
9.49  696 

9.49  74o 
9.49  784 
9.49  828 

o.5o  526 
o.5o48i 
o.5o  437 

o.5o  393 
o.5o348 
o.5o  3o4 

o.5o  260 
o.5o  216 
o.5o  172 

9.97978 
9.97  974 
9.97970 

9.97966 
9.97962 
9.97  958 

9.97954 
9.97950 
9.97  946 

39 
38 
37 
36 
35 
34 

33 

32 

3i 

30 

9.47  8i4 

40 

9- 

49872 

o.5o  128 

9.97  942 

30 

L.  Cos. 

d. 

L. 

Cotg.     d. 

L.  Tang. 

L.  Sin. 

d. 

' 

72°  3O  . 

PP 

.1 

.2 

-3 

•4 

1 

•9 

45 

44            43 

.2 
•3 

•4 

42 

41 

40 

4                3 

4-5 
9.0 

13-5 

18.0 
22.5 
27.0 

3r-5 
36.0 

40.5 

tl      *i 

13.2          12.9 
17.6          17.2 

22.0              21-5 
26.4              25.8 

30.8              30.1 

35-2          34-4 

8.4 

12.6 

16.8 

21.  0 
25-2 

29.4 

33-6 

8.2 

12.3 

16.4 

20.5 

24.6 

28.7 

32.8 

8.0 

12.  0 

16.0 
20.  o 

24.0 
28.0 

32.0 

36.0 

.2 

•  3 

•4 
•  5 
.6 

•  7 
.8 

0.8            0.6 
1.2            0.9 

1.6                   1.2 
2.O                   1-5 

2.4             1.8 

2.8                  2.1 
3-2                  2.4 

3-6            2.7 

64 


17°  30 


J 

L.  Sin. 

d. 

L 

.  Tang.     d. 

L.  Cotg. 

L.  Cos. 

d. 

30 

9.47  8i4 

9 

.49  872 

o.5o  128 

9-97 

942 

30 

3i 

9.47 

854 

9 

.49  916 

44 

o.5o  o84 

9-97 

938 

29 

32 

9.47 

894 

9 

.49  960 

o.5o  o4o 

9-97 

934 

28 

33 

9-47 

934 

40 

9 

.  5o  oo4 

44 

0.49  996 

9-97 

4 

27 

34 
35 
36 

9.47 
9-48 
9-48 

974 
014 
o54 

40 
40 

9 
9 
9 

.5oo48 
.  5o  092 
.5o  1  36 

44 
44 
44 

0.49  952 
0.49  908 
0.49  864 

9.97926 
9.97922 
9.97918 

4 
4 
4 

26 

25 
24 

37 
38 
39 

9-48 
9-48 
9.48 

094 
i33 
i73 

40 

39 
40 

9 
9 
9 

.5o  180 

.50  223 

.  5o  267 

44 
43 
44 

0.49  820 
0.49  777 
0.49  733 

9-97 
9-97 
9-97 

910 
906 

4 
4 

4 

23 
22 
21 

40 

9-48 

2l3 

9 

.5o3n 

0.49  689 

9-97 

902 

20 

4i 

9.48 

252 

39 

9 

.5o355 

44 

0.49  645 

9-97 

898 

19 

42 

9-48 

292 

9 

.5o398 

0.49  602 

9-97 

8o4 

18 

43 

9.48 

332 

4° 

9 

.  5o  442 

44 

0.49558 

9-97 

890 

'7 

44 

9-48 

37i 

39 

9 

.5o485 

43 

0.49  5i*j 

9-97 

886 

4 

16 

45 

9-48 

4u 

9 

.5o  529 

0.49  471 

9-97 

882 

i5 

46 

9-48 

45o 

39 

9 

.5o  572 

43 

0.49  428 

9-97 

878 

i4 

4° 

44 

4 

47 

9.48  490 

39 

9 

.5o6i6 

0.49  384 

9.97874 

4 

i3 

48 

9.48 

529 

9 

.5o659 

0.49  34i 

9-97 

870 

12 

49 

9-48 

568 

39 

9 

.5o  703 

44 

0.49  297 

9-97 

866 

I  I 

50 

9.48 

607 

39 

9 

.50746 

43 

0.49  254 

9-97 

861 

10 

5i 

9-48 

647 

9 

.5o  789 

43 

0.49  211 

9-97 

857 

9 

52 

9-48 

686 

9 

5o833 

O.49  167 

9-97 

853 

8 

53 

9.48 

725 

39 

9 

.50876 

43 

O.49  124 

9-97 

849 

7 

54 

9-48 

764 

39 

9 

.5o  919 

43 

0.49  08  I 

9-97 

845 

4 

6 

55 

9-48 

8o3 

9 

.  5o  962 

0.49  o38 

9-97 

84  1 

5 

56 

9-48 

842 

39 

9 

.  5  1  oo5 

43 

0.48  995 

9-97 

837 

4 

57 

9-48 

881 

39 

9 

.5i  o48 

43 

0.48952 

9-97 

833 

4 
4 

3 

58 

9-48 

920 

9 

.5l   092 

o.48  908 

9-97 

820 

2 

59 

9-48 

959 

39 

9 

.5i  135 

43 

0.48865 

9-97 

825 

I 

60 

g.48 

998 

39 

9 

.5i  178 

43 

o.48  822 

9-97 

821 

0 

L.  Cos.      d. 

L 

.  Cotg.      d.     L.  Tang. 

L.  Sin.     d. 

f 

72° 

PP 

44 

43 

42 

* 

40 

39 

5               4 

, 

4-4 

4-3 

4.2 

.1 

4.1 

4.0 

3-9 

.1 

0.5            0.4 

.2 

8.8 

8.6 

8.4 

.2 

8.2 

8.0 

7.8 

.2 

i.o            0.8 

•3 

13.2 

12.9 

12.6 

•3 

12.3 

I2.O 

11.7 

•  3 

1-5                  1-2 

•4 

17.6 

17.2 

16.8 

•4 

,6.4 

16.0 

15-6 

•4 

2.0                 1.6 

•5 

22.  0 

21-5 

21.  0 

•5 

20.5 

20.  o 

*9*  5 

•5 

2.5                 2.0 

.6 

26.4 

25-8 

25-2 

.6 

24.6 

24.0 

23-4 

.6 

3.0                 2.4 

•7 

30.8 

30.1 

29.4 

•  7 

28.7 

28.0 

27-3 

•  7 

3-5            2.8 

.8 

35-2 

34-4 

33-6 

.8 

32-8 

32.0 

31.2 

.8 

4.0            3.2 

9        39-6 

38-7        37-  8 

.9 

36.0 

35-  x 

4-5             3-6 

65 


18C 


L.  Sin. 

!  d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

0 

9.48  998 

9-5i  178 

40 

0.48  822 

9 

97821 

60 

I 

9.49  037 

39 

9-51   221 

43 

0.48  779 

9 

97817 

59 

2 

9.49  076 

9.61  264 

o.48  736 

9 

97812 

58 

3 

9.49  115 

39 
38 

9.  5  1  3o6 

43 

o.48  694 

9 

97  808 

4 

57 

4 

9.49  i53 

9.5i  349 

43 

o.4865i 

9 

97  8o4 

56 

5 

9.49  192 

9.5i  392 

o.48  608 

9 

97  800 

55 

6 

9.49  23i 

39 

9.5i4 

*5 

o.48  565 

9 

97796 

4 

54 

38 

43 

4 

7 

9.49  269 

9.51478 

42 

0.48  522 

9 

97792 

53 

8 

9.49  3o8 

9.5i  5- 

20 

o.4848o 

9 

97788 

52 

9 

9.49  347 

39 

9.5i  563 

43 

0.48437 

9 

97784 

4 

5i 

10 

9.49  385 

JC 

9.  5  1  606 

o.48  3g4 

9 

97779 

50 

1  1 

9.49  424 

39 

9.61  648 

0.48352 

9 

97775 

49 

12 

9.49  462 

3 

9.5i  691 

o.48  309 

9 

97  771 

48 

i3 

9.49  5oo 

38 

9.5i  734 

43 

o.48  266 

9 

97  767 

4 

47 

i4 

9.49  539 

39 

,0 

9.61  776 

42 

0.48  224 

9 

97763 

4 

46 

i5 

9.49  577 

3° 

9-5i  819 

0.48  181 

9 

97  759 

45 

16 

9.49  6  1  5 

38 

9-5i  861 

42 

o.48  139 

9 

97754 

5 

44 

17 

9.49654 

39 

-0 

9-5i  903 

42 

0.48  097 

9 

9775o 

4 

43 

id 

9.49692 

9.  5  1  946 

o.48o54 

9 

97  746 

42 

19 

9.49  73o 

38 

9.5i  9! 

48 

42 

O.48  012 

9 

97  742 

4i 

20 

9.49  768 

38 

9.52  o3i 

43 

0.47  969 

9 

97738 

40 

21 

9.49806 

38 

-0 

9.52  073 

42 

0.47  927 

9 

97734 

4 

39 

22 

9.49844 

3° 

9.52  1  15 

0.47885 

9 

97  729 

38 

23 

9.49882 

38 

9.52  157 

42 

0.47843 

9 

97725 

4 

37 

24 

9.49920 

38 

_0 

9.52  200 

43 
42 

O.47  8OO 

9 

97  721 

4 

36 

25 

9.49958 

3° 

9.52  242 

0.47758 

9 

97  7*7 

35 

26 

9.49996 

38 

9.52  284 

42 

0.47  716 

9 

97  7i3 

4 

34 

27 

9.5o  o3^ 

38 

9.52  326 

42 

42 

0.47  674 

9 

97708 

5 

33 

28 

9.5o  072 

3° 

9.52  368 

0.47  632 

9 

97  7°4 

32 

29 

9.5o  1  10 

38 

9.524 

10 

42 

0.47  590 

9 

97  700 

4 

3i 

30 

9.5o  i48 

38 

9.52  452 

0.47548 

9 

97696 

4 

30 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L.  Sin. 

d. 

71°  3D. 

PP 

43 

42 

39 

38 

5 

4 

.1 

4-3 

4.2 

.T 

3-9 

3-8 

.1 

o-5 

0.4 

.2 

8.6 

8.4 

.2 

7.8 

7.6 

.2 

I.O 

o.tf 

•3 

12.9 

12.6 

•  3 

11.7 

11.4 

•3 

i-5 

1.2 

•4 

17.2 

16.8 

•4 

15-6 

15-2 

•4 

2.0 

1.6 

•5 

21.5 

21.0 

•  5 

IQ-  5 

19.0 

.  5 

2-5 

2.0 

.6 

25-8 

25.2 

.6 

23-4 

22.8 

.6 

2-4 

•7 

30-1 

29.4 

•7 

27-3 

26.6 

•7 

3-5 

2.8 

.8 

34-4 

33-6 

.8 

31.2 

30.4 

.8 

4.0 

3-2 

•9 

38.7 

37-8 

35-  l               34-2 

•9 

4-5                 3-6 

66 


18°  3O 


L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

30 

9.  5o  i48 

9.52  452 

o. 

47548 

9.97696 

30 

3i 

9.5o  i85 

38 

9.52 

494 

42 

o. 

47506 

9-97 

69I 

29 

32 

9.  5o  223 

,g 

9.52 

536 

0. 

47  464 

9.97  687 

28 

33 

9-5o  261 

9.52 

578 

o. 

47422 

9-97 

683 

4 

27 

37 

42 

- 

34 

9«5o  298 

38 

9.52 

620 

41 

o. 

4738o 

9-97 

679 

26 

35 

9.5o336 

18 

9.52 

661 

o. 

47339 

9.97674 

25 

36 

9-5o  374 

9.52 

7o3 

o. 

47297 

9-97 

670 

4 

24 

37 

42 

37 

9.5o4n 

38 

9.52 

745 

42 

0. 

47255 

9-97 

666 

23 

38 

9-5o  44 

9 

9.52 

787 

0. 

9-97 

662 

22 

39 

9-5o486 

37 

9.52 

829 

42 

o. 

47  171 

9-97 

657 

b 

21 

40 

9.5o  523 

37 

9.52 

870 

0. 

47  i3o 

9-97 

653 

20 

4i 

9.5o  56i 

38 

9.52 

912 

0.47  088 

9-97 

649 

19 

42 

9.5o  598 

9.52 

953 

o. 

47  047 

9-97 

645 

18 

43 

9.5o635 

37 

9.52  995 

42 

0.47  005 

9-97 

64o 

5 

'7 

44 

9.5o  673 

38 

9.53  037 

42 

o.46  963 

9.97636 

4 

16 

45 

9-5o  710 

9.53078 

o  46  922 

9-97 

632 

i5 

46 

9-5o  747 

37 

9.53 

I2O 

42 

o. 

4688o 

9-97 

628 

i4 

47 

9.5o  784 

37 

9.53 

161 

41 

o. 

46839 

9-97 

623 

5 

i3 

48 

9«5o  821 

37 

9.53 

2O2 

o. 

46798 

9.97619 

1 

12 

49 

9-5o858 

37 

9.53 

244 

42 

o. 

46756 

9-97 

6,5 

4 

II 

50 

9.50896 

38 

9.53 

285 

41 

o.46  715 

9-97 

610 

10 

5i 

9.5o933 

37 

9.53 

327 

42 

o. 

46673 

9-97 

606 

1 

9 

52 

9.5o  970 

37 

9.53 

368 

o. 

46632 

9-97 

602 

8 

53 

9-5i  007 

37 

9.53  409 

41 

0. 

465gi 

9-97 

597 

5 

7 

54 

9.  5  1  o43 

36 

9.5345o 

41 

0. 

46550 

9-97 

593 

4 

6 

55 

9.  5  1  080 

37 

9.53 

492 

o. 

465o8 

9-97 

589 

5 

56 

9.5i  117 

37 

9.53 

533 

41 

o. 

46  467 

9-97 

584 

• 

4 

57 

9.5i  i54 

37 

9.53 

574 

41 

o. 

46426 

9-97 

58o 

4 

3 

58 

9-5i  191 

37 

9.53 

6i5 

o. 

46385 

9-97 

576 

2 

59 

9-5i  227 

36 

9.53 

656 

4i 

0. 

46  344 

9-97 

671 

I 

60 

9.  5  1  264 

37 

9.53 

697 

4i 

o. 

46  3o3 

9-97 

567 

0 

L.  Cos. 

d. 

L.  Cotg. 

d.  j  L. 

Tang. 

L.  Sin. 

d. 

71°. 

PP       42 

4i 

38 

37 

36 

5 

4 

.1        4.2 

4.1 

3-8 

•i            3-7 

3-6 

.! 

0.5 

04 

8.4 

8.  a 

7.6 

.2           7.4 

7-2 

.2 

l.O 

0.8 

•  3         I2-6 

12.3 

11.4 

3          "-1 

10.8 

•3 

i-5 

1.2 

.4         16.8 

16.4 

15-2 

.4          14.8 

14.4 

•4 

2.O 

1.6 

•5             21.0 

20.5 

19.0 

•5          18.5 

18.0 

2-5 

2.0 

.6         25.2 

24.6 

22.8 

.6             22.2 

21.6 

.6 

3-° 

2.4 

•7         29.4 

28.7 

26.6 

•7          25.9 

25.2 

•7 

3-5 

2.8 

.8         33-6 

32.8 

30.4 

.8         29.6 

28.8 

.8 

4.0 

3-2 

36.9         34.2 

32-4 

4-5 

67 


19°. 


, 

L.  Sin. 

d. 

L.  Tang. 

d. 

L 

.  Cotg. 

L.  Cos.     d. 

0 

9.5i  264 

9.53  697 

0 

46  3o3 

9-97 

567 

60 

I 

9.5i  3oi 

37 

9.53738 

o 

46  262 

9-97 

563 

5 

59 

2 

9.5i  338 

,6 

9.53779 

O.46  221 

9-97 

558 

58 

3 

9.5i  374 

37 

9.  53  820 

41 

0 

46  180 

9-97 

554 

4 
4 

57 

4 

9.5i  4i 

I 

36 

9.53  861 

o 

46  i39 

9-97 

550 

56 

5 

9.  5  1  447 

9.53  902 

o 

46  098 

9-97 

545 

55 

6 

9.5i  484 

36 

9.53943 

41 
41 

o.46  057 

9-97 

54  1 

4 

5 

54 

7 

9.5i  52O 

37 

9.53984 

o. 

46  016 

9-97 

536 

53 

8 

9.5i  557 

9.54  025 

o 

45  975 

9-97 

532 

52 

9 

9.5i  593 

Jj 

9.54 

o65 

40 

0. 

45  935 

9-97 

528 

4 

5i 

10 

9.5i  629 

9.54  1  06 

0. 

45894 

9-97 

523 

50 

ii 

9.5i  666 

•*6 

9.54  147 

41 

o. 

45853 

9-97 

5i9 

4 

49 

12 

9.5i  702 

9-54 

187 

o. 

458i3 

9-97 

515 

48 

i3 

9.5i  738 

30 
36 

9-54 

228 

41 

0. 

45  772 

9-97 

Sio 

b 
4 

47 

i4 

9.5i  774 

37 

9-54 

269 

0. 

4573i 

9-97 

5o6 

46 

i5 

9.5i  81 

I 

9-54 

309 

0. 

45  691 

9-97 

5oi 

45 

16 

9.5i  847 

36 

9-54 

350 

41 

0. 

45  65o 

9-97 

497 

4 

44 

'7 

9.5i  883 

,6 

9-54 

390 

40 

o. 

456io 

9-97 

492 

b 

43 

18 

9.5i  91 

q 

9-54 

43i 

0. 

45  569 

9-97 

488 

42 

J9 

9.5i95 

5 

36 

9-54 

471 

4° 

0. 

45  529 

9-97 

484 

4 

4i 

20 

9.5i99i 

3° 

9-54 

5l2 

41 

0. 

45488 

9-97479 

40 

21 

9.52  027 

36 

9-54 

552 

4° 

o. 

45448 

9-97475 

4 

39 

22 

9.52  o63 

9-54 

593 

0. 

45407 

9-97 

47o 

38 

23 

9.52  099 

3° 

9.54633 

40 

0. 

45  367 

9-97 

466 

4 

37 

36 

4° 

5 

24 

9.52  i3 

5 

16 

9-54 

673 

0. 

45  327 

9-97 

46i 

36 

25 

9.52  171 

9-54 

7i4 

0. 

45  286 

9-97 

457 

35 

26 

9.52  207 

36 

9-54 

4° 

o. 

45  246 

9-97 

453 

4 

34 

27 

9.52  242 

35 
16 

9-54 

794 

40 

0. 

45  206 

9-97 

448 

5 

33 

28 

9.52  278 

9-54 

83,5 

0. 

45  i65 

9-97 

444 

32 

29 

9.52  3i4 

36 

9.54 

875 

40 

o. 

45  125 

9-97 

439 

b 

3i 

30 

9.52  350 

36 

9.54 

915 

4° 

0. 

45  o85 

9-97 

435 

30 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L. 

Tang. 

L.  Sin. 

d. 

7O°  3D  . 

PF 

>         41 

40 

37 

36 

35 

5               4 

.1 

4-1 

4.0 

3-7 

3-6 

3-5 

.1 

0.5            0.4 

.2 

8.2 

8.0 

7-4 

.2               7.2 

7.0 

.2 

i.o           0.8 

•3 

12.3 

12.0 

n.  i 

.3         10.8 

10.5 

•3 

1-5                  !-2 

•4 

16.4 

16.0 

14.8 

•4         14-4 

14.0 

•4 

2.0                 1.6 

•5 

20.5 

20.0 

18.5 

.5         18.0 

17-5 

•5 

2.5                  2.0 

.6 

24.6 

24.0 

22.2 

.6         21.6 

21.  0 

-      .6 

3.0                 2.4 

•7 

28.7 

28.0 

25.9 

•7         25.2 

24-5 

•7 

3-5             2.8 

.8 

32.8 

32.0 

20.6 

.8         28.8 

28.0 

.8 

4-0             3-2 

•9         36-9 

36.0               33.3 

•9          32-4 

31-5 

•9 

4-5             3-6 

68 


19°  3D . 


t 

L.  Sin. 

d. 

L.  Tang. 

d. 

L 

,  Cotg. 

L.  Cos. 

d. 

30 

9.52  350 

35 
36 
35 
36 
35 
36 
35 
36 
35 
36 

35 
35 
36 
35 
35 
35 
35 
35 
35 

9-54 

9*5 

40 
4o 

40 
40 
4o 
40 
40 
40 
4o 
40 

40 
40 

39 
40 
40 
40 

39 
40 
40 
39 
40 

39 
40 

39 
40 

39 
40 

39 
39 
40 

o. 

45o85 

9.97435 

5 
4 
5 
4 
5 
4 
5 
4 
5 

30 

3i 

32 

33 

34 
35 
36 

37 

38 

39 

9.52  385 
9.52  421 
9.52456 

9.52  492 
9.52  527 
9.52  563 

9.52  598 
9.52634 
9.52  669 

9.54905 
9.54  995 
9.55  o35 

9-55  075 
9.55  n5 
9.55  i55 

9.55  i95 
9.  55  235 
9.55  275 

0. 

o. 
o. 

0. 

o. 
o. 

0. 

o. 
o. 

45  045 
45  005 
44  965 

44925 
44885 
44845 

44805 
44765 
44  725 

9.97430 
9.97  426 
9.97421 

9.97417 
9.97  412 
9.97  4o8 

9.97  4o3 
9.97  399 
9.97  394 

29 
28 
27 

26 

25 
24 

23 
22 
21 

40 

9.52  705 

9.55 

315 

o. 

44685 

9-97 

390 

20 

4i 

42 

43 

44 

45 
46 

47 
48 

49 

9.52  740 
9.52  775 
9.52  811 

9.52  846 
9.52  881 
9.52  916 

9.52  gSi 
9.52  986 

9.53  021 

9-55355 
9.55395 
9.55434 

9.55474 
9.555i4 
9.55554 

9.55593 
9.55633 
9.55673 

0.44645 
o.44  6o5 
0.44566 

o.44  526 
0.44486 
0.44446 

o.44  407 
o.44  367 
o.44  327 

9-97 
9-97 
9-97 

9-97 
9-97 
9-97 

9-97 

9-97 
9-97 

385 
38i 
376 

372 
367 
363 

358 
353 
349 

4 
5 
4 
5 
4 
5 
5 
4 

«9 
1  8 

17 

16 
i5 
i4 
i3 

12 
I  I 

50 

9-53  o56 

35 
-e. 

9.55 

712 

o. 

44288 

9-97 

344 

10 

Si 

52 

53 

54 
55 
56 

57 
58 
59 

9.53  092 
9.53  126 
9.53  161 

9-53  196 
9.53  23i 
9-53  266 

9.53  3oi 
9.53  336 
9.53  37o 

3° 
34 
35 
35 
35 
35 
35 
35 
34 

9.55  752 
9-55  791 
9.  5583i 

9-55  870 
9.  55  910 
9-55  949 

9.55  989 
9.56  028 
9.56  067 

o.44  248 
o.44  209 
0.44  169 

o.44  i3o 
o.44  090 
o.44  o5  1 

0.44  on 
o.43  972 
o.43  933 

9-97 
9-97 
9-97 

9-97 
9-97 
9-97 

9-97 
9-97 
9-97 

34o 
335 
33i 

326 

322 

3i7 

3l2 

3o8 
3o3 

5 
4 

5 
4 

5 
5 
4 

5 

9 

8 

7 
6 
5 
4 
3 

2 

60 

9.534o5 

35 

9.56 

107 

o.43893 

9-97 

299 

0 

L.Cos.       d. 

L.  Cotg. 

d. 

L. 

Tang. 

L.  Sin. 

d. 

/ 

70°. 

PP       40 

39 

36 

35 

34 

.1 

.2 

•3 
•4 

:I 

:l 

•9 

5               4 

.1          4.0 

.2              8.0 

•  3        12.0 

.4        16.0 

•  5            20.0 

.6        24.0 

.7        28.0 
.8        32.0 
.g         36.0 

1:1 
11.7 

15-6 
19-5 
23-4 

27-3 
31.2 
35.  i 

3-6 
7.2 
10.8 

14.4 
18.0 

21.6 

25.2 
28.8 

32.4 

•i          3-5 
.2          7.0 
•3        i°-5 

.4        14.0 
•5         17-5 

.6            21.0 

•7         24.5 
.8        28.0 
•9         3T-5 

u 

10.2 

13.6 
I7.0 
20.4 

23-8 
27.2 
30.6 

0.5            0.4 
i.o            0.8 
1.5            i-2    * 

2.0                  1.6 
2-5                 2.0 
3-0                 2.4 

3-5            2.8 
4-0           3-2 

4-5            3-6 

69 


2O°. 


L.  Sin. 

d. 

L.  Tang.     d. 

L. 

Cotg. 

L.  Cos.     d. 

0 

9.534o5 

9.56 

I07 

0. 

43  893 

9-97 

299 

60 

I 

9.53440 

35 

9.56 

i46 

39 

o. 

43854 

9.97  294 

5 

59 

2 

9-53475 

9.56 

i8b 

0. 

438i5 

9-97 

289 

58 

3 

9.53  509 

34 

9.56 

224 

39 

o. 

43776 

9-97 

285 

4 

57 

35 

4° 

4 

9.53544 

9.56 

264 

39 

o. 

43736 

9-97 

280 

56 

5 

9.53578 

9.56 

3o3 

o. 

43697 

9-97 

276 

55 

6 

9.536i3 

35 

9.56 

342 

39 

o. 

43658 

9-97 

271 

b 

54 

7 

9.53  647 

34 

9.56 

38i 

39 
39 

0.43  619 

9-97 

266 

5 

53 

8 

9.53682 

9.56 

420 

o. 

4358o 

9-97 

262 

52 

9 

9.53  716 

34 

9.56 

459 

39 

o. 

4354i 

9-97 

25? 

5 

5i 

10 

9.5375i 

35 

9.56 

498 

39 

0. 

43  5o2 

9-97 

252 

50 

1  1 

9.53  785 

9.56 

537 

39 

o. 

43463 

9-97 

248 

4 

49 

12 

9.53819 

9.56 

576 

o. 

43424 

9-97 

243 

' 

48 

i3 

9.53854 

35 

9.56 

6i5 

o. 

43385 

9-97 

238. 

b 

47 

34 

39 

i4 

9.53888 

34 

9.56 

654 

39 

o. 

43  346 

9-97 

234 

46 

i5 

9-53  922 

9.56 

693 

o. 

433o7 

9-97 

229 

' 

45 

16 

9.53957 

35 

9.56 

732 

0. 

43268 

9-97 

224 

- 

44 

34 

39 

»7 

9.53991 

34 

9.56771 

39 

o. 

43  229 

9-97 

220 

43 

18 

9.54  025 

9.56 

810 

o. 

43  190 

9-97 

2l5 

42 

19 

9.54  059 

34 

9.56 

849 

39 
,a 

o. 

43  i5i 

9-97 

210 

- 

4i 

20 

9.54093 

34 

9.56 

887 

o  . 

43  n3 

9-97 

2O6 

4 

40 

21 

9.54  127 

34 

9.56 

926 

39 

o. 

43o74 

9-97 

2OI 

5 

39 

22 

9.54  161 

9.56 

965 

o. 

43o35 

9-97 

196 

38 

23 

9.54  195 

34 
34 

9.57 

oo4 

18 

0. 

42996 

9-97 

192 

4 
5 

37 

24 

9.54  229 

9.57 

042 

39 

o. 

42958 

9-97 

l87 

36 

25 

9.54263 

9.57 

08  1 

o. 

42  919 

9-97 

182 

35 

26 

9.54297 

34 

9.57 

I2O 

39 

o. 

42  880 

9-97 

178 

4 

34 

27 

9.5433i 

34 

9.57 

i58 

3° 
39 

o. 

42  842 

9-97 

I73 

) 

33 

38 

9.54365 

9.57 

197 

_Q 

o. 

42803 

9-97 

1  68 

32 

29 

9.54399 

34 

9.57 

235 

3° 

o. 

42765 

9-97 

i63 

b 

3i 

30 

9.54433 

34 

9.57 

274 

o. 

42  726 

9-97 

i59 

30 

L.  Cos. 

d. 

L.  Cotg.      d. 

L.  Tang. 

L.Sin. 

d. 

' 

69°  3O  . 

PP         40 

39 

38 

35 

34 

5 

4 

I         4.0 

3-9 

3-8 

•i            3-5 

3-4 

., 

0.5 

0.4 

2                 8.0 

7.8 

7.6 

.2            7.0 

6.8 

.2 

I.O 

0.8 

3          12.0 

11.7 

11.4 

•3          JQ-S 

IO.  2 

•3 

i-5 

1.2 

4          16.0 

15-6 

1-5-2 

.4          14.0 

I3.6 

•4 

2.0 

1.6 

5          20.  o 

19-5 

19.0 

•5          17-5 

I7.0 

•5 

2-5 

2.0 

6          24.0 

23-4 

22.8 

.6              21.  0 

20-4 

.6 

3-o 

2-4 

7          28.0 

27-3 

26.6 

•7          24.5 

238 

•7 

3-5 

2.8 

8         32.0 

31.2 

3°-4 

.8          28.0 

27.2 

.8 

4.0 

3-2 

Q          36-0 

34.2 

•9           V-S 

4-5 

3-6 

70 


2O°  3O 


' 

L.  Sin.       d. 

L.  Tang.     d. 

L. 

Cotg. 

L.  Cos. 

d. 

30 

9.54433 

9.57 

274         38 

0. 

42  726 

9-97 

i59 

30 

3i 

9.54466 

34 

9.57 

3i2       M 

o. 

42688 

9-97 

1  54 

5 

29 

32 

9.54  5oo 

9.57 

35i 

o. 

42  649 

9-97 

i4g 

28 

33 

9.54534 

34 

9.57 

389 

38 

0.42  611 

8.97 

145 

4 

27 

34 

9.54567 

33 

34 

9.57428 

39 
38 

o. 

42  572 

9-97 

i4o 

5 

26 

35 

9.54601 

9.57466 

0. 

42  534 

9-97 

i35 

25 

36 

9.54635 

34 

9.57 

5o4 

38 

o. 

42496 

9-97 

i3o 

b 

24 

3? 

9-54668 

33 

9.57  543 

39 
-,% 

0.42457 

9-97 

126 

4 

23 

38 

9.  54  702 

9.57 

58i 

o. 

42  4i9 

9-97 

121 

22 

39 

9.54735 

33 

9.57 

619 

38 

0. 

42  38  1 

9-97 

116 

s 

21 

40 

9-54  769 

34 

9.57 

658 

08 

0.42  342 

9-97 

in 

20 

4i 

9.54802 

33 

9.57 

696 

3° 

og 

o. 

42  3o4 

9-97 

107 

4 

i9 

42 

9-54836 

9.57 

734 

0. 

42266 

9-97 

102 

1  8- 

43 

9.54869 

33 

9.57 

772 

38 

0. 

42  228 

9-97 

°97 

b 

17 

44 

45 

9.54903 
9.54936 

34 
33 

9.57  810 
9-57  849 

38 
39 

0. 

o. 

42   I90 

42  i5i 

9-97092 
9-97  °87 

5 

5 

16 
i5 

46 

9.54969 

33 

9.57 

887 

38 

o. 

42  n3 

9.97o83 

4 

i4 

4? 

9-55  oo3 

34 

9.57925 

38 
18 

o. 

42  075 

9-97 

078 

5 

i3 

48 

9-55o36 

9.57963 

o. 

42o37 

9-97 

o73 

12 

49 

9.55  069 

33 

9-58  ooi 

38 

o. 

4i  999 

9-97 

068 

b 

I  I 

50 

9-55   102 

33 

9.58  039 

38 

o. 

4i  96i 

9-97 

o63 

5 

10 

5i 

9.55  i36 

34 

9.58077 

38 
18 

0. 

4i  923 

9-97 

o59 

9 

52 

9.55  169 

9.58 

"J 

o.4i  885 

9-97 

o54 

8 

53 

9-55  202 

33 

9.58 

i53 

38 

0. 

41847 

9.97o49 

S 

7 

54 

9-55235 

33 

9.  58 

191 

38 
18 

o. 

4  1  809 

9-97 

o44 

5 

6 

55 

9.55  268 

9.58 

229 

0. 

4i  771 

9-97 

o39 

5 

56 

9.55  3oi 

33 

9-58 

267 

38 

o. 

4i  733 

9-97035 

4 

4 

5? 

9.55334 

33 

9-58 

3o4 

37 

0. 

4  1  696 

9-97 

o3o 

5 

3 

58 

9.55  367 

33 

9.58 

342 

3° 

o. 

4i  658 

9-97 

°25 

2 

59 

9.55  4oo 

33 

9.58 

38o 

38 

0. 

4i  620 

9-97 

020 

5 

I 

60 

9.55433 

33 

9.584i8 

S8 

o. 

4i  582 

9-97 

oi5 

0 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L. 

Tang. 

L.Sin. 

d. 

' 

69°. 

PP          39 

38 

37 

34 

33 

5               4 

•i            3-9 

3-8 

3-7 

.!                 3.4 

3-3 

.1 

0.5            0.4    . 

7-8 

7.6 

7-4 

.2           6.8 

6.6 

.2 

i.o            0.8 

•3          "-7 

11.4 

u.  i 

•  3               IO.  2 

9-9 

•3 

1-5                  1-2 

•4          »5-6 

15-2 

14.8 

•4          13-6 

13.2 

•4 

2.0                   1.6 

•5          19-5 

10.  0 

18.5 

•5          i7-° 

16.5 

•  5 

2.5                 2.O 

.6          23.4 

22.8 

22.2 

.6         20.4 

19.8 

6 

3-0                 2.4 

•7          27-3 

26.6 

25-9 

•7          23-8 

23.1 

7 

3-5            2.8 

.8          31.2 

3°-4 

29.  6 

.8         27.2 

26.4 

.8 

4.0            3.2 

•9          35-  * 

34-2           33-3 

•  9          30-6 

29.7 

•9 

4-5             3-6 

71 


21°. 


1 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

0 

9.55433 

33 
33 
33 
32 
33 
33 
33 
32 
33 

9.584i8 

37 
38 
38 
38 
37 
38 
37 
38 
38 
37 
38 
37 
38 
37 
37 
38 
37 
38 
37 
37 

37 

38 
37 
37 
37 
37 
38 
37 
37 
37 

o.4i  582 

9.97  oi5 

5 
5 
4 
5 
5 
5 
5 
5 
5 
5 

4 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
4 
5 
5 
5 
5 
5 
5 
5 

60 

I 

2 

3 

4 
5 
6 

7 

8 

9 

9.55466 
9.55499 
9.55532 

9.  55  564 
9.55597 
9.55  63o 

9.55663 
9.55695 
9.55  728 

9.58  455 
9.58493 
9.  5853i 

9.58  569 
9.58  606 
9.58  644 

9.58  681 
9  58  719 
9.58757 

o.4i  545 
o.4i  507 
o.4i  469 

o.4i  43i 
o.4i  394 
o.4r  356 

o.4i  319 
o.4i  281 

o.4i  243 

9.97  oio 
9.97  oo5 
9.97  ooi 

9.96996 
9.96991 
9.96  986 

9.96  981 
9.96  976 
9.96971 

59 

58 

56 
55 
54 
53 

52 

5i 

10 

9-55  761 

9-58  794 

0.4  1  206 

9.96  966 

50 

ii 

12 

i3 

i4 

i5 

16 

17 

18 

'9 

9-55  793 
9.55826 
9.55858 

9-55  891 
9.55  923 
9.55956 

9.55988 
9.56  02  1 
9.56o53 

33 
32 
33 
32 
33 
32 
33 
32 
32 

33 
32 
32 
33 

32 
32 
32 
32 
32 
33 

9.58832 
9.58869 
9-58  907 

9.58944 
9.58  981 
9.59  019 

9.59  o56 
9.59  094 
9.59  i3i 

o.4i  168 
o.4i  i3i 
o.4i  093 

o.4i  o56 
o.4i  019 
o.4o  981 

o.4o  944 
o.4o  906 
o.4o  869 

9.96  962 
9.96957 
9.96  952 

9.96947 
9.96  942 
9.96937 

9.96  932 
9.96927 
9  .  96  922 

49 

48 

47 

46 
45 
44 

43 

42 

4i 

20 

9-56o85 

9.59  168 

o.4o  832 

9.96917 

40 

21 
22 
23 

24 
25 
26 

27 
28 
29 

9,56  118 
9.56  iScf 
9-56  182 

9.56  215 
9.56  247 
9.56  279 

9.  563u 
9.  56  343 
9.56  375 

9.59  2o5 
9.59  243 
9.59  280 

9-59  317 
9.59  354 
9.59  391 

9.59  429 
9.59  466 
9.59  5o3 

o.4o  795 
o.4o  757 
o.4o  720 

o.4o683 
o.4o646 
o.4o  609 

o.4o  571 
o.4o534 
o.4o  497 

9.96  912 
9.96907 
9.96  903 

9.96  898 
9.96  893 
9.96  888 

9.96  883 
9.96  878 
9.96  873 

39 
38 
37 

36 
35 
34 
33 

32 

3i 

30 

9.56  4o8 

9.59^540 

o.4o  46o 

9.96  868 

30 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L.  Sin. 

d. 

68°  30'. 

PP 

.1 

.2 

•3 

•4 

•  7 
.8 

— 

38 

37 

33 

32 

5 

4 

11.4 
15-2 

19  o 

22.8 

26.6 
3°-4 

34-2 

3-7 
7-4 
n.  i 

14.8 
18.5 

22.2 

25-9 
29.6 

.  i 

.2 

•  3 

•4 
•  5 
.6 

' 

U 

9.9 

13.2 
16.5 
19.8 

23.1 
26.4 

3-2 

6.4 
9.6 

12.8 

16.0 
19.2 

22.4 

25.6 

.1           0.5 

.2                1.0 

•3            i-5 

.4                2.0 

•5           2.5 

•7            3-5 
.8           4.0 

•9             4-S 

0.4 
0.8 

1.2 

1.6 

2.0 
2.4 

2.8 

72 


21°  3O 


L.  Sin. 

d. 

L.  Tang.     d. 

L.  Cotg. 

L.  Cos. 

d. 

30 

9.564o8 

32 
32 

32 
32 
32 

31 
32 
32 
32 

9.  59  54o 

37 
37 
37 
37 
37 
37 
37 
36 
37 
37 
37 
37 
36 
37 
37 
37 
36 
37 
37 
36 

37 
36 

37 
36 
37 
36 
37 
36 
37 
36 

o.4o  46o 

9 

96  868 

5 
5 

5 
5 
5 
5 
5 
5 
5 
5 

5 

5 
5 
5 
5 
5 
5 
5 
6 

5 

5 
5 
5 
5 
5 
5 
5 
5 
5 
5 

30 

3i 

32 

33 

34 
35 
36 

38 
39 

9-56  44o 
9.56  472 
9.565o4 

9.56536 
9.56  568 
9.  56  599 

9.  5663i 
9.56663 
9-56  695 

9.59  577 
9.59  6i4 
9.59  65i 

9.59  688 

9-59  725 
9.59  762 

9.59799 
9.59835 
9.59  872 

0.40  423 
o.4o  386 
o.4o  349 

o.4o  3i2 
o.4o  275 
o.4o238 

O.4o  2OI 

o.4o  165 
o.4o  128 

9.96  863 
9.96  858 
9.96853 

9.96848 
9.96  843 
9.96  838 

9.96833 
9.96  828 
9.96  823 

29 

28 
27 

26 

25 

24 

23 
22 
21 

40 

9.56  727 

9.59909 

o.4o  091 

9 

96  818 

20 

4i 

42 

43 

44 
45 
46 

47 
48 

49 

9.56  759 
9-56  790 
9.56822 

9.56854 
9.56886 
9.56917 

9-j>6  949 
9.56  980 

9.57  012 

S2 
32 
32 

32 
32 

9.59  946 
9.59  983 
9  .60  019 

9-6oo56 

9.60  i3o 

9.60  166 
9.60  2o3 
9.60  240 

o.4o  o54 
o.4o  017 
0.39  981 

0.39  944 
0.39  907 
0.39  870 

0.39834 
o.  39797 
0.39  760 

9.96  8i3 
9.96  808 
9.96  8o3 

9.96  798 
9.96793 
9.96  788 

9.96  783 
9.96  778 
9.96772 

'9 

18 

17 
16 
i5 
i4 
i3 

12 
I  I 

50 

9.57  o44 

9.60  276 

0.39  724 

9- 

96  767 

10 

5i 

52 

53 

54 
55 
56 

57 

58 

59 

9.57075 
9.57  107 
9.57  i38 

9.57  169 

9.57  2OI 
9.57  232 

9,57  264 
9.57  326 

32 

32 
3' 
32 

3' 

9.60  3  1  3 
9.60  349 
9.60  386 

9.60  422 
9.60  459 
9.60  495 

9.60  532 
9.60  568 
9.60  605 

0.39  687 
0.39  65i 
0.39  6i4 

0.39  578 
0.39  54  1 
0.39  505 

0.39468 
0.39  432 

0.39  395 

000  OOO  000 

96  762 
96  757 
96  752 

96747 
96  742 
96737 

96  732 

96  727 
96  722 

9 

8 

7 
6 
5 
4 
3 

2 
I 

60 

9.57358 

9  .60  64  1 

0.39  359 

9- 

96  717 

0 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L.Sin.     d. 

68°. 

PP 

.1 

.2 

•3 
•4 

•9 

37 

36 

32 

31 

2 

3 
4 

I 

•  9 

6 

5 

3-7 
7-4 
n.  i 

14.8 
18.5 

22.2 

25-9 
29.6 

33-3 

3-6 
,o'.8 

21  6 

25  2 
288 
32.4 

.1 

.2 

•  3 
•4 

:78 

12.8 

16.0 
19.2 

22.4 
25.6 

K 

9-3 

12.4 
15-5 
18.6 

21.7 
24.8 

27.9 

06 

I  2 

i  8 
24 

t! 

5-4 

0.5 

I.O 

2.0 
2-5 

3-5 
4.0 

4-5 

/ 

L.  Sin. 

d. 

L. 

Tang. 

d. 

L.  Cotg. 

L.  Cos.  d. 

0 

9.57 

358 

9.  60  64  1 

o.  3g 

359 

9.96 

60 

I 

9.57389 

31 

9.6o  677 

37 

0.39 

323 

9.96 

711 

59 

2 

9.57  420 

9.60  714 

,6 

0.39 

286 

9.96 

706 

58 

3 

9.5745i 

31 

9.60  750 

36 

o.Sg 

25o 

9-96 

7oi 

5 

S 

57 

4 

9.57  482 

32 

9.60  786 

37 

0.39 

214 

9.96 

696 

56 

5 

9-57 

5i4 

9- 

60823 

,5 

o.  39 

177 

9.96 

691 

55 

6 

9  57 

545 

31 
31 

9.60  85g 

36 

o.  39 

i4i 

9.96686 

5 
5 

54 

7 

9.57 

576 

9- 

60  895 

36 

o.  3g 

105 

9.96 

681 

53 

8 

9  .  57  607 

9- 

60931 

16 

0.39 

069 

9.96 

676 

52 

9 

9.57638 

31 

9- 

60  967 

3° 

0.39 

o33 

9.96 

670 

5i 

10 

9.57  669 

3r 

9- 

6  1  oo4 

36 

o.38 

996 

9.96 

665 

50 

1  1 

9-57 

7oo 

31 

9- 

6  1  o4o 

36 

o.38 

960 

9.96 

660 

49 

12 

9-57 

73i 

9- 

61  076 

o.38 

924 

9.96 

655 

48 

i3 

9.57 

762 

31 

9- 

61112 

36 

o,38 

888 

9.96 

650 

5 
5 

47 

i4 

9.57 

793 

31 

9- 

61  1  48 

36 

o.38 

852 

9.96 

645 

46 

i5 

9.57 

824 

9- 

61  1  84 

,6 

o.38 

816 

9.96 

64o 

45 

16 

9  57 

855 

31 

3° 

9- 

6l  220 

36 

o.38 

780 

9.96 

634 

S 

44 

17 

9.  57  885 

31 

9- 

6r  256 

36 

o.38 

744 

9.96 

629 

43 

18 

9.57 

916 

9- 

61  292 

16 

o.38 

708 

9.96 

624 

42 

r9 

9.57947 

9- 

61  328 

06 

o.38 

672 

9.96 

619 

4i 

20 

9.57 

978 

9- 

61  364 

16 

0.38 

636 

9.96 

6i4 

40 

21 

9.58  008 

3° 

9- 

6  1  4oo 

36 

o.38 

600 

9.96  608 

39 

22 

9.58 

o39 

9- 

61  436 

o.38 

564 

9.96 

6o3 

38 

23 

9.58 

070 

31 

9.61  472 

3° 
36 

o.38 

528 

9.96 

598 

b 
5 

37 

24 

9.58 

IOI 

9.61  5o8 

16 

o.38 

492 

9.96 

593 

36 

25 

9.58 

i3i 

9.61  544 

o.38 

456 

9.96 

588 

35 

26 

9.58 

162 

31 

9,61  579 

35 

o.38 

421 

9.96 

582 

34 

27 

9.58 

192 

30 

9.61  6i5 

36 

o.38 

385 

9.96 

577 

5 

33 

28 

9.58 

223 

9.61  65i 

o.38 

349 

9.96 

572 

32 

29 

9.58 

253 

30 

9.61  687 

36 

o.38 

3i3 

9,96 

567 

b 

3r 

30 

9.58  284 

31 

9.61  722 

35 

o.38 

278 

9.96 

562 

5 

30 

L.  Cos. 

d. 

L. 

Cotg. 

d. 

L.  Tang. 

L.  Sin. 

d. 

67°  3O  . 

PP 

37 

36     35 

32 

31 

3° 

6      5 

.1 

3-7 

3-6     3  5 

, 

3-2 

I'1 

3° 

I 

o.  6     o.  5 

7-4 

7-2     70 

.2 

6.4 

6.2 

6.0 

2 

12       1.0 

•3 

u.  i 

10  8     10,5 

•3 

9.6 

9-3 

9.0 

3 

1.8     1.5 

•4 

14.8 

14-4     14-0 

4 

12.8 

12.4 

12.  0 

4 

2  4       2.0 

:i 

18.5 

22.2 

18.0    17.5 

21.6      21.0 

•  5 
6 

16.0 
19.2 

III 

15.0 

18.0 

.1 

3.0       2.5 

36     3-0 

•  7 

25-9 

25.2     24  5 

,7 

22  4 

21  7 

21.  O 

•  7 

42     3-5 

.8 

29.6 

28.8     280 

,8 

25-6 

248 

24.0 

.8 

4.8     4.0 

9    33  3 

32.4     31  5 

•9 

28.8    27.9 

9 

5-4     4-5 

74 


22°  3O 


, 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

30 

9- 

58  284 

9.61  722 

o.38  278 

9.96  562 

6 

30 

3i 

9.583i4 

3° 

9.61  758 

3° 
76 

o.38  242 

9.  96  556 

s 

29 

32 

9 

58  345 

9.61  ' 

r94 

o.38  206 

9.96  55i 

28 

33 

9 

58  375 

3° 

9.61  83o 

36 

o.38  170 

9.96  546 

27 

34 

9 

58  4o6 

3' 

9.61  865 

35 
_< 

o.38  135 

9.96  54  1 

5 
6 

26 

35 

9 

58436 

9  61  901 

3 

o.  38  099 

9.96535 

25 

36 

9 

58467 

3' 

9.61  936 

35 

o.38  o64 

9>96  53o 

24 

37 

9 

58497 

30 

9,61  972 

36 

3* 

o.38  028 

9.96525 

23 

38 

9 

58  527 

9.62  008 

0.37992 

9.96  52O 

6 

22 

39 

9 

58557 

3° 

9.62  o43 

35 

0.37  957 

9.96  5i4 

5 

21 

40 

9 

58588 

31 

9.62  079 

3° 

0.37  921 

9.96  5o9 

5 

20 

4i 

9 

.586i8 

3° 

9.62  u4 

35 
06 

o.37  886 

9.96  5o4 

6 

19 

42 

9 

58648 

9,62 

50 

0.37  85o 

9.96  498 

18 

43 

9 

.58  678 

3° 

9.62  i85 

35 

0.37815 

9.96  493 

17 

44 

9 

.58  709 

3' 

9.62  221 

36 

0.37  779 

9.96488 

5 

5 

16 

45 

9 

.58739 

30 

9.62  256 

0.37  744 

9.96483 

6 

i5 

46 

9 

.58  769 

30 

9.62  292 

36 

0.37  708 

9.96477 

i4 

47 

9 

.58799 

30 

9.62  327 

35 

o.37673 

9.96472 

i3 

48 

9 

.58  829 

9.62  362 

o.37  638 

9.96  467 

fi 

12 

49 

9 

.58859 

3° 

9.62  398 

36 

0.37  602 

9.96  46  1 

5 

I  I 

50 

9 

.58  889 

3° 

9.62  433 

o.37567 

9.96  456 

10 

5i 

9 

.  58  9i9 

3° 

9.62  468 

35 
16 

0.37  532 

9.96 

45i 

6 

9 

52 

9 

.58949 

3° 

9.  62  5o4 

0.37  496 

9.96  445 

8 

53 

9 

.58979 

30 

9.62 

539 

35 

0.37461 

9.96  44o 

7 

54 

g 

.  59  oo9 

30 

9.62 

574 

35 

0.37426 

9.96 

435 

6 

6 

55 

g 

.59  o3^ 

I 

3° 

9.62  609 

0.37  391 

9.96 

429 

5 

56 

9 

.59  o69 

3° 

9.62 

645 

36 

o.37  355 

9-96 

424 

4 

57 

58 

9 
9 

.59  o98 
.5o  128 

29 
30 

9.62 
9.62 

680 

35 
35 

0.37  320 

o.37285 

9.96  4i9 
9.96  4i3 

6. 

3 

2 

59 

9.59  i  58 

3° 

9.62 

75o 

35 

0.37  250 

9.96  4o8 

5 

I 

60 

9 

.59  188 

3° 

9.62 

785 

35 

o. 

37215 

9.96  4o3 

0 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L. 

Tang. 

L.  Sin. 

d. 

' 

67°. 

PP 

36 

35 

31 

30 

29 

6 

5 

.2 

3.6 
7.2 

3-5 
7.0 

K 

.1 

2 

3-° 
6.0 

!;§ 

.1 
.2 

0.6 

1.2 

o-S 

I.O 

•3 

10.8 

10.5 

9-3 

•3 

9.0 

8.7 

•3 

1.8 

15 

•4 

14.4 

14.0 

12.4 

4 

12.  0 

it  6 

•4 

2.4 

2.0 

18.0 

21.6 

21  0 

SI 

1 

15-0 

18.0 

M-5 
17.4 

:1 

3-° 
3-6 

25 

.7 

25.2 

24-5 

21.7 

•7 

21.0 

20.3 

•7 

4-2 

3-5 

.8 

28.8 

28.0 

24.8 

.8 
g 

24.0 
27.0 

£ 

.8 

•9 

4.8 

40 

23°. 


' 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos.     d. 

0 

9.69  i»8 

3° 
29 

3° 
3° 
29 

30 
3° 
29 
30 
29 

3<> 
29 
3° 
29 
30 
29 
29 
3° 
29 
29 
30 
29 
29 

29 
29 

3° 
29 
29 

29 
29 

9.62 

785 

35 
35 
35 
36 
35 
35 
35 
35 
35 
34 

35 
35 
35 
35 
35 
35 
34 
35 
35 
35 
35 
34 
35 
35 
34 
35 
34 
35 
35 
34 

0. 

37215 

9.96  4o3 

6 
5 
5 
6 

5 
6 

5 
5 
6 
5 
6 
5 
5 
6 

5 
6 
5 
6 
5 
6 

5 
5 
6 
5 
6 
5 
6 

5 
6 
5 

60 

I 

2 

3 

4 
5 
6 

7 
8 

9 

9.69  218 
9.69  247 
9.69277 

9.59  807 
9.59  336 
9.59  366 

9.59  396 

9.59  425 
9.59455 

9.62  820 
9.62  855 
9.62  890 

9.62  926 
9.62  961 
9.62  996 

9.63  o3i 
9.63  066 
9-63  101 

o. 
o. 

0. 
0. 

o. 

0. 

o. 
o. 

0. 

37  180 
37  145. 
37  1  10 

37074 

37o39 
37  oo4 

36  969 
36934 
36899 

9.96 

9.96 
9.96 

9.96 
9.96 
9.96 

9.96 
9.96 
9.96 

397 
392 
387 

38i 
376 
37o 

365 
36o 

354 

59 
58 
57 
56 
55 
54 

53 

52 

5i 

10 

9.59  484 

9.63 

i35 

0. 

36865 

9.96 

349 

50 

1  1 

12 

i3 

i4 
i5 
16 

'7 

18 

'9 

9.59  5i4 
9.59543 
9.59573 

9.59  602 
9.59  632 
9.59661 

9.59  690 
9.59  720 
9.59749 

9-63  170 
9.63  2o5 
9.63  24o 

9.63  275 
9«63  3io 
9.  63  345 
9.63379 
9.634i4 
9.63  449 

0. 

o. 
o. 

o. 
o. 

0. 

o. 
o. 

0. 

3683o 

36795 
36  760 

36725 
36  69o 
36655 

36621 
36586 
36  55i 

9.96 
9.96 
9.96 

9.96 
9.96 
9.96 

9.96 
9.96 
9.96 

343 
338 
333 

327 
322 

3i6 

3ii 
3o5 
3oo 

49 
48 

47 

46 

45 
44 

43 

42 

4i 

20 

9.59778 

9.63 

484 

0. 

365i6 

9.96 

294 

40 

21 
22 
23 

24 
25 

26 

27 
28 
29 

9.59808 
9.59837 
9.59866 

9.59895 
9.59924 
9.59  954 

9.59  983 
9.60  012 
9.60  o4i 

9.63 
9.63 
9.63 

9.63 
9.63 
9.63 

9.63 
9.63 
9.63 

5i9 
553 

588 

623 
657 
692 

726- 
761 

796 

o. 
o. 
o. 

o. 
o. 
o. 

o. 
•o. 

0. 

3648i 
36447 
364i2 

36377 
36  343 
363o8 

36274 
36239 
362o4 

9.96  289 
9.96  284 
9.96  278 

9.96  273 
9.  96  267 
9.96  262 

9.96  256 
9.96  25i 
9.96  245 

39 

38 

37 
36 
35 
34 

33 

32 

3i 

30 

9.  60  070 

9.63 

83o 

0. 

36  170 

9.96 

240 

30 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L. 

Tang. 

L.  Sin.     d. 

' 

66°  30  . 

PP       36 

35 

34 

3° 

29 

.1 

.2 

•3 
•4 

! 

:l 

— 

6                5 

•i            3-6 

.2                  7.2 

.3         10.8 

•4          M-4 
.5          18.0 
.6         21.6 

•7         25-2 
.8         28.8 

.q           ^2.4 

3-5 
7.0 
10.5 

14.0 
I7-S 

2I.O 

24-5 
28.0 

31.5 

3-4 
6.8 

IO.2 

13-6 
17.0 
20.4 

23.8 
27.2 

' 

3.0 

2            6.0 
3            9-° 

.4              12.0 

•  s        15-0 

.6          18.0 

.7              21.0 

,8          24.0 
•9          27-° 

2.9 
5-8 
8.7 

n.6 
M-5 
17.4 

20.3 

a: 

0.6            0.5 

1.2                  1.0 

1.8            1.5 

2.4                 2.O 
3.0                 2-5 
3.6                 3.0 

4-2            3-5 
4.8            4.0 

5-  4             4-5 

76 


23°  3O 


' 

L.  Sin. 

d. 

L.  Tang.     d. 

L.  Cotg. 

L. 

Cos. 

d. 

30 

9 

.60  070 

9.  6383o 

35 
34 
35 
34 
35 
34 
35 
34 
34 
35 

34 
34 
35 
34 
34 
35 
34 
34 
34 
34 

35 
34 
34 
34 
34 
34 
34 
34 
34 
34 

o.36  170 

9.96  240 

30 

3i 

32 

33 

34 
35 
36 

3? 

38 
39 

9 
9 
9 

9 
9 

9 

9 
9 
9 

.60  o99 
.60  128 
.60  157 

.60  186 
.60  215 
.60  244 

.60273 
.60  3o2 
.6o33i 

29 

29 
29 

29 

29 

29 
29 
29 

9.  63  865 
9.63899 
9.63  934 

9.  63  968 
9.64oo3 
9.64  037 

9.64  072 
9.64  1  06 
9.64  i4o 

o.36i35 
o.36  101 
o.36  066 

o.36o32 
o.35997 
o.35  963 

o.35  928 
o.35894 
o.35  860 

9.96  234 
9.96  229 

9.96  223 

9.96  2  1  8 

9.96  212 
9.96  207 

9.96  2O1 
9.96  196 
9.96  190 

5 
6 
5 
6 

5 
6 
5 
6 
5 

29 
28 
27 

26 

25 
24 

23 
22 
21 

40 

9.6o359 

9.  64  175 

0.35825 

9-96  185 

20 

4i 

42 

43 

44 
45 
46 

47 
48 

49 

9 
9 
9 

9 
9 
9 

9 
9 
9 

.60  388 
.60  417 
.60  446 

.60474 
.6o5o3 
.60  532 

,6o56i 
.6o589 
60618 

29 

29 
29 
28 
29 
29 

29 
28 
29 

9.64  209 
9.64243 
9.64  278 

9.64  3i2 
9.64346 
9.6438i 

9.64415 
9-64449 
9.64483 

o.35  791 
0.35757 
o.35  722 

0.35688 
0.35654 
o.35  619 

0.35585 
o.3555i 
o.355i7 

9.96179 
9.96  174 

9.96  168 

9.96  162 
9.96  i57 
9.96i5i 

9.96  1  46 
9.96  i4o 
9.96135 

5 
6 
6 
5 
6 
5 
6 

5 
g 

19 

17 

16 
i5 
i4 

i3 

12 
I  I 

50 

9 

60  646 

29 
29 
28 
29 
28 
29 
.  28 
.  29 
28 

9.64  5  1  7 

0.35483 

9.96  i29 

10 

5r 

52 

54 
55 
56 

57' 
58 
59 

9 
9 
9 

9 
9 
9 

9 

9 

9 

6o675 
60  704 
.60  732 

60  761 
60  789 
60818 

.6o846 
.60875 

9.64  552 
9.64586 
9.64  620 

9.64654 
9.64688 
9.64  722 

9.64756 
9.64  790 
9.64  824 

0.35448 
o.354i4 
o.3538o 

0.35346 
o.35  3i2 
o.35  278 

0.35244 
o.35  210 
o.35  176 

9.96  123 
9.96  118 
9.96  112 

9.96  107 
9.96  101 
9.96  095 

9.96  090 
9.96  o84 
9.96079 

5 
6 

5 
6 
6 
5 
6 
5 

1 

7 
6 
5 
4 

3 

2 

I 

60 

9 

.60  93i 

28 

9.  64  858 

o.35  142 

9.96073 

0 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L. 

Sin. 

d. 

' 

66°. 

PP 

.2 

•3 
•4 

•J 

>9 

35 

34 

n 

28 

.2 

3 
•4 

•9 

6 

5 

3-5 

7.0 
10.5 

14.0 
J7  5 

21.0 

24-5 
28.0 

31.5 

U 

IO.  2 

13-6 
17.0 
20.4 

23-8 
27.2 

3O.O 

,1 

.2 

3 

•4 

•Jt 

•9 

tl 

n.6 
17.4 

20.3 
23.2 

26.1 

2.8 

5-6 
8.4 

11.2 
14.0 

16.8 

19.6 
22.4 

0.6 

1.2 

1.8 
2.4 

tJ 

5-4 

0-5 

I.O 

2.O 
2-5 

3-5 
4.0 

77 


24C 


' 

L.  Sin. 

d. 

L.  Tang. 

d. 

L. 

Cotg. 

L.  Cos. 

d. 

0 

9.60  931 

9.64 

858 

0. 

35  142 

9.96 

o73 

60 

I 

9.60  960 

28 

9-64 

892 

34 

o. 

35  108 

9.96 

067 

59 

2 

9.60  988 

28 

9-64 

926 

o. 

35o74 

9.96 

062 

58 

3 

9.61  016 

9-64 

960 

34 

o. 

35  o4o 

9.96 

o56 

57 

29 

34 

6 

4 

9.61  045 

28 

9-64 

994 

0. 

35  006 

9.96 

o5o 

56 

5 

9.61  o73 

_0 

9.  65 

028 

o. 

34972 

9.96 

045 

55 

6 

9.61  10 

i 

9.65 

062 

34 

o. 

34938 

9.96 

o39 

54 

7 

9.61  129 

29 

9.  65 

096 

34 

o. 

34  904 

9.96 

o34 

5 

6 

53 

8 

9.61  i58 

9.65 

i3o 

o. 

3487o 

9.96 

028 

52 

9 

9.61  186 

28 

9.65 

1  64 

34 

0. 

34836 

9.96 

022 

5i 

10 

9.61  214 

9.  65 

197 

33 

o. 

348o3 

9.96 

OI7 

50 

ii 

9.61  242 

28 

9.65 

23l 

34 

0. 

34769 

9.96 

Ol  I 

6 

49 

12 

9.61  270 

9.65 

265 

o. 

34  73<) 

9.96 

oo5 

48 

i3 

9.61  298 

28 

9.65 

299 

34 

o. 

34  701 

9.96 

ooo 

5 

47 

i4 

9.61  326 

28 

28 

9.65 

333 

34 

0. 

34667 

9.95 

994 

fi 

46 

i5 

9.61  354 

9.  65 

366 

33 

o. 

34634 

9-95 

988 

45 

1  6 

9.61  382 

28 

9.  65 

4oo 

34 

o. 

34  600 

9.95 

982 

44 

I? 

9.61  4i 

i 

29 

9.  65 

434 

34 

0. 

34  566 

9.95 

977 

6 

43 

18 

9.61  438 

9.  65 

467 

33 

o. 

34533 

9.95 

971 

42 

'9 

9.61  466 

9.65 

5oi 

34 

o. 

34499 

9.95 

965 

4i 

20 

9.61  494 

9.65 

535 

34 

o. 

34465 

9.95 

960 

5 
g 

40 

21 

9.61  522 

28 

9.65 

568 

33 

0. 

34432 

9.96 

954 

6 

39 

22 

9.61  55o 

9-65 

602 

34 

o. 

34398 

948 

38 

23 

9.61  578 

28 

9.  65 

636 

34 

o. 

34364 

9.95 

942 

37 

24 

9.61  606 

28 

28 

9.65 

669 

33 

o. 

3433i 

9.95 

937 

b 
6 

36 

25 

9.61  634 

9.65 

7o3 

34 

o. 

34  297 

9.95 

93i 

5 

35 

26 

9.61  662 

28 

9.65 

736 

33 

o. 

34  264 

9.95 

925 

34 

27 

9.61  689 

27 

28 

9.65 

770 

34 

o. 

3423o 

9.95 

920 

b 
6 

33 

28 

9.61  717 

9-65 

8o3 

33 

0. 

34  197 

9.95 

914 

g 

32 

29 

9.61  745 

28 

9.65 

837 

34 

o. 

34  i63 

9.95 

908 

6 

3i 

30 

9.61  773 

9.65 

87o 

33 

o  . 

34  i3o 

).95 

902 

30 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L. 

Tang. 

L.  Sin.  |d. 

65°  30  . 

PP    34 

33 

29 

28 

27 

6      5 

•i     3-4 

3-3 

2.9 

.1       2.8 

2.7 

.1 

0.6     0.5 

.2     6.8 

66 

5-8 

2     5-6 

54 

.2 

1.2        1.0 

•3    10.2 

9-9 

8.7 

•3     8-4 

8.1 

•3 

1.8     1.5 

•4    13-6 

13.2 

11.6 

.4      II.  2 

10.8 

•4 

2.4       2.0 

.5    17.0 

16.5   ' 

14-5 

.5      14.0 

J3-5 

•5 

3.0       2.5 

.6    20.4 

19.8 

17.4 

.6    16.8 

16.2 

.6 

3-6-     3-o 

•7    238 

23.1 

20.3 

.7    19.6 

18.9 

7 

42     3-5 

.8    27.2 

26.4 

23.2 

.8    22.4 

21.6 

.8 

4.8     4.0 

.9    30.6 

29.7     26.1 

•  9    25.2 

24-3 

5-4     4-5 

78 


24°  3D' 


> 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

30 

9 

61  773 

27 

9.65  870 

34 

o.  34  i3o 

9.95  902 

30 

3i 

9 

61  800 

28 

9.65  90^ 

33 

o.34  o96 

9.95  897 

6 

29 

32 

9 

61  828 

28 

9.65  937 

o.34  o63 

9.  95  891 

28 

33 

9 

61  856 

27 

9.65  97i 

33 

O.34  O29 

9.95885 

6 

27 

34 

9.6i  883 

28 

9.66  oo4 

34 

0.33  996 

9.95  879 

26 

35 

9 

61  9n 

28 

9.66  o38 

o.33  962 

9.95873 

25 

36 

9 

61  939 

27 

9.66  071 

33 

o.33  929 

9.95  868 

6 

24 

37 

9 

61  966 

28 

9.66  10^ 

34 

o.33  896 

9.95  862 

6 

23 

38 

9 

61  994 

9.  66  1  38 

o.33  862 

9.95  856 

22 

39 

9 

62  021 

27 

28 

9.66  171 

33 

o.33  829 

9.95  85o 

5 

21 

40 

9 

62  o49 

9.66  204 

o.33  796 

9.95  844 

20 

4i 

9 

62  076 

28 

9.  66  238 

34 

o.33  762 

9.95  839 

6 

,9 

42 

9 

62  io4 

9.66  271 

o.33  729 

9.95  833 

18 

43 

9 

62  i3i 

28 

9.66  3o4 

33 
33 

o.33  696 

9.95  827 

6 

17 

44 

9 

62  i59 

27 

9.66  337 

0.33663 

9.95  821 

g 

16 

45 

9 

62  186 

28 

9.66  371 

o.33  629 

9.95  8i5 

i5 

46 

9 

62  214 

9.66  4o4 

33 

o.33  596 

9.95  810 

5 

i4 

27 

33 

47 

9 

62  241 

27 

9.  66  437 

o.33  563 

9.95  8o4 

f, 

i3 

48 

9 

62  268 

9.66  470 

33 

o.33  53o 

9.95  798 

12 

49 

9 

62  296 

9.665o3 

33 

0.33497 

9.95792 

6 

II 

50 

9 

62  323 

9.66  537 

34 

0.33463 

9.95  786 

6 

10 

5i 

9 

62  35o 

27 

9.66  570 

33 
33 

o.3343o 

9.95  780 

s 

9 

52 

9 

62377 

9.66  6o3 

o.33  397 

9-95  775 

g 

8 

53 

9 

62  405 

9.  66  636 

33 

o.33  364 

9.95  769 

7 

27 

33 

54 

9 

62432 

9.66  669 

o.33  33i 

9.95763 

6 

6 

55 

9 

62  459 

9.66  702 

o.33  298 

9.95  757 

6 

5 

56 

9 

62486 

27 

9.66  735 

33 

o.33  265 

9.95  75i 

4 

57 

9 

62  5i3 

27 
28 

9.66  768 

33 

o.33  232 

9.95  745 

6 

3 

58 

9 

62  54  1 

9.66  801 

o.33  199 

9-9 

5  739 

5 

2 

59 

9 

62  568 

27 

9.  66  834 

33 

o.33  166 

9.95733 

I 

60 

9 

62  595 

27 

9.66  867 

33 

o.33  1  33 

9.95  728 

0 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L. 

Sin. 

d. 

65°. 

PP 

34 

33 

28 

27 

6                5 

, 

3-4 

3-3                   -i 

2.8 

2-7 

., 

0.6              0.5 

.2 

6.8 

6.6                   .2 

5.6 

5-4 

.2 

1-2                     I.O 

•3 

IO.  2 

9-9                   3 

8.4 

•3 

1.8              1.5 

-4 

I3.6 

13-2                   -4 

II   2 

0.8 

•4 

2-4                      2.0 

-5 

17.0 

16.5                    5 

14.0 

3-5 

.  tj 

3.0                      2.5 

.6 

20-4 

19.8                  .6 

16.8 

6.2 

6 

3-6              3-° 

•  7 

23-8 

23-1                   -7 

19,6 

8.9 

.7 

4-2              3-5 

.8 

27.2 

26.4                    8 

22.4 

1.6 

8 

4.8              4.0 

25.2              4.3 

•9 

5-4               4-5 

79 


25°. 


/ 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.Cotg. 

L. 

Cos.     d. 

0 

9 

.62695 

9.66  867 

o.33  i33 

9.96  728 

g 

60 

I 

9 

.62  622 

27 

9.66  900 

33 

o.33  100 

9.96  722 

6 

59 

2 

9 

.62  649 

9.66933 

0.33067 

9.96  716 

58 

3 

9 

.62676 

27 

9.66  966 

33 

o.33  o34 

9.96  710 

6 

57 

4 

9 

.62  703 

27 

9.66999 

33 

o.33  ooi 

9.96  704 

6 

56 

5 

9 

.62  730 

9.67  o32 

0.32  968 

9.96  698 

55 

6 

9 

.62767 

9.67  065 

o.32  935 

9.96  692 

54 

27 

33 

6 

7 

9 

.62  784 

9.67  098 

33 

o.32  902 

9.96  686 

6 

53 

8 

9 

.62  811 

9.67i3 

[ 

0.32  869 

9.96  680 

52 

9 

9 

.62838 

27 

9.67  i63 

32 

o.32837 

9.96  674 

5i 

10 

9 

.62  865 

9.67  196 

33 

o.32  8o4 

9.96  668 

50 

1  1 

9 

.62  892 

26 

9.67  229 

33 

o.32  771 

9.96  663 

6 

49 

12 

9 

.62  918 

9.67  262 

o.32  738 

9.96  667 

48 

i3 

9 

.62  945 

27 

9.67295 

32 

o.32  706 

9.96  65i 

6 

47 

i4 

9 

.62  972 

27 

9.67327 

33 

o.32  673 

9-95645 

6 

46 

16 

9 

.62  999 

9.67  36o 

o.32  64o 

9.96  639 

45 

16 

9 

.63  026 

27 
26 

9.6739 

3 

33 

o.32  607 

9.96  633 

6 

44 

17 

9 

.63  062 

27 

9.67  42< 

5 

32 

o.32  674 

9.96  627 

fi 

43 

18 

9 

.63  079 

9.67458 

0.32  542 

9.96  621 

42 

'9 

9 

.63  106 

27 

9.67  491 

33 

o.32  609 

9.96615 

4i 

20 

9 

.63  i33 

„/: 

9.67  624 

o.32  476 

9.96  609 

40 

21 

9 

.63  169 

27 

9.67  556 

33 

o.32  444 

9.96  6o3 

6 

39 

22 

9 

.63  186 

9.67  689 

o.32  4 

[  I 

9.96  697 

38 

23 

9 

.63  2i3 

27 
26 

9.67  622 

33 
32 

o.32  378 

9.96691 

6 

37 

24 

9 

.63  239 

9.67  654 

33 

o.32  346 

9.96685 

36 

26 

9 

.63266 

9.67  687 

o.32  3i3 

9.96  679 

35 

26 

9 

.63  292 

9.67  719 

32 

0.32  28l 

9.96673 

34 

27 

33 

6 

27 

9 

.633i9 

26 

9.67  762 

0.32  248 

9.96667 

6 

33 

28 

9 

.63345 

9.67785 

0.32  2l5 

9.96  56i 

32 

29 

9 

.63  372 

27 

9.67  817 

32 

o.32  i83 

9.95555 

3i 

30 

9 

.63  398 

9.67  850 

33 

o.32  i5o 

9.96  549 

30 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L. 

Sin.     d. 

64°  30  . 

PP 

33 

32 

37 

26 

6 

5 

i 

3-3 

3-2 

.    .1 

2-7 

2.6 

,  I 

0.6 

o-5 

2 

6.6 

6.4 

.2 

5-4 

5-2 

.2 

1.2 

I.O 

3 

9-9 

9.6 

•3 

8.1 

7.8 

•3 

1.8 

i-5 

4 

13.2 

12.8 

•4 

10.8 

10.4 

•4 

2-4 

2.O 

5 

16.5 

16.0 

•  5 

13-5 

13.0 

•  5 

3-° 

2-5 

6 

19.8 

19.2 

6 

16.2 

15.6 

.6 

3-6 

3-° 

7 

23-1 

22.4 

.7 

18.9 

18.2 

•7 

4.2 

3-5 

8 

26.4 

25-6 

.8 

21.6 

20.8 

.8 

4.8 

4.0 

Q 

24.3           23.4 

S-4                4-5 

80 


25°  3D 


' 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

30 

9 

63  398 

9.67  850 

o.32  i5o 

9.95  549 

30 

3i 

9 

63425 

26 

9.67  882 

32 

o.32  118 

9.95543 

29 

32 

9 

6345i 

9.67  915 

33 

o.32  o85 

9.95537 

28 

33 

9 

63478 

27 

9.67947 

32 

o.32  o53 

9.9553i 

6 

27 

34 
35 

9 
9 

63  53i 

26 
27 

9.67-980 

9.68  012 

33 
32 

O.32  O2O 

o.3i  988 

9.95525 
9.95  5i9 

6 
6 

26 

25 

36 

9 

63  557 

9.68044 

32 

o.3i  956 

3  5i3 

6 

24 

3? 

9 

63583 

26 

9.68  077 

33 

o.3i  923 

9.95  507 

6 

23 

38 

9 

636io 

9.68   109 

o.3i  891 

9.95  5oo 

22 

39 

9 

63636 

26 

9.68  142 

33 

o.3i  858 

9.95  494 

6 

21 

40 

9 

63  662 

9.68  174 

32 

o.3i  826 

9.95488 

20 

4i 

9 

63  689 

26 

9.68  206 

33 

o.3  1  794 

9.95  482 

6 

I9 

42 

9 

63  715 

26 

9.68  239 

o.3i  761 

9.95  476 

18 

43 

9 

6374i 

9.68  271 

o.3i  729 

9.95470 

17 

26 

32 

6 

44 

9 

63  767 

27 

9.68  3o3 

o.3i  697 

9.95464 

6 

16 

45 

9 

63794 

9.68  336 

o.3i  664 

9.95458 

i5 

46 

9 

63820 

26 

9.68  368 

32 

o.3i  632 

9.95  452 

6 

i4 

47 

9 

63846 

26 

9.68  4oo 

o.3i  600 

9.95  446 

6 

i3 

48 

9 

63872 

9.68  432 

32 

o.3i  568 

9.95  44o 

12 

49 

9 

63898 

9.68  465 

33 

o.3i  535 

9.95434 

II 

50 

9 

63  924 

26 

9.68  497 

32 

o.3i  5o3 

9.95427 

10 

5i 

9 

6395o 

26 

9.68  529 

o.3i  471 

9.95  421 

6 

9 

52 

9.  63  976 

26 

9.  6856i 

o.3i  439 

9.95  4i5 

8 

53 

9.64  002 

26 

9.68593 

32 
33 

o.3i  407 

9.95  4oQ 

6 

7 

54 

9 

64  028 

26 

9.68  626 

o.3i  374 

9.95  4o3 

6 

6 

55 

9 

64o54 

9.68  658 

o.3i  342 

9.95  397 

5 

56 

9 

.64  080 

9.68  690 

32 

o.3i  3io 

9.95  39i 

4 

5? 

9 

.64  1  06 

26 
26 

9.68  722 

32 

o.3i  278 

9.95  384 

7 
6 

3 

58 

9 

.64  1  32 

9.68  754 

o.3i  246 

9.95378 

2 

59 

9 

.64  i58 

9.68  786 

32 

o.3i  214 

9.95372 

g 

I 

60 

9 

.64  1  84 

9.68  818 

32 

o.3i  182 

9.95  366 

0 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L. 

Sin. 

d. 

' 

64°. 

PP 

33 

32 

27 

26 

7 

6 

.1 

3-3 

3-2 

., 

2.7 

2.6                           .1 

0.7 

0.6 

.2 

6.6 

6.4 

.2 

5-4 

5-2                           -2 

M 

1.2 

•3 

9.9 

9.6 

•3 

8.1 

7-8                   -3 

2.1 

1.8 

•4 

13.2 

12.8 

•4 

10.8 

10.4                   .4 

2.8 

2.4 

.5 

16.5 

16.0 

!3-5 

i3-°                    5 

3-5 

3.0 

.6 

19.8 

19.2 

.6 

16.2 

15.6                   .6 

4.2 

3-6 

.7 

23.1 

22.4 

•7 

18.9 

18.2                   .7 

4-9 

4.2 

.8 

26.4 

25.6 

.8 

21.6 

20.8                   .8 

5-6 

4.8 

•9 

29.7             28.8 

•9 

24-3            23-4                   -9 

6.3               5-4 

81 


26°. 


/ 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L. 

Cos. 

d. 

0 

9.64  1  84 

26 

9.68  818 

o.3i  182 

9.95  366 

60 

I 

9.64  210 

26 

9.68  85o 

32 

o.3i  150 

9.95  36o 

6 

59 

2 

9.  64  236 

26 

9.68  882 

o.3i  118 

9.95354 

58 

3 

9.64  262 

26 

9.68  914 

32 

32 

o.3i  086 

9.95  348 

7 

57 

4 

9.64  288 

25 

9.68  946 

o.3i  o54 

9.95  34i 

6 

56 

5 

9.64  3i3 

26 

9.68  978 

o.3i  022 

9.95335 

55 

6 

9.64  339 

26 

9.69  oio 

32 

32 

o.3o  990 

9.95  329 

6 

54 

7 

9 

.64365 

26 

9.69  042 

0.30958 

9.95  323 

6 

53 

8 

9 

.6439i 

26 

9.69074 

o.3o  926 

9.95  3i7 

52 

9 

9 

.64417 

9.69  106 

32 

o.3o  894 

9.95  3io 

7 

5i 

10 

9 

.64442 

26 

9.69  1  38 

32 

o.  3o862 

9.95  3o4 

50 

ii 

9 

.64468 

26 

9.69  170 

32 

o.3o  83o 

9.95  298 

6 

49 

12 

9 

.64494 

9.69  202 

o.3o  798 

9.95  292 

48 

i3 

9 

.645i9 

25 

9.69  234 

32 

o.3o  766 

9.95  286 

47 

i4 

9 

.64545 

26 
26 

9.69  266 

32 

o.3o  734 

9.95  279 

7 

6 

46 

i5 

9 

.6457i 

9.69  298 

o.3o  702 

9.95  273 

45 

16 

9 

.64  596 

25 

9.69  329 

31 

o.3o  671 

9.95  267 

44 

17 

9 

.64622 

26 

9.69  36i 

32 

o.3o  6 

39 

9.95  261 

b 

43 

18 

9 

.64647 

9.6939 

3 

32 

o.3o  607 

9.95  254 

42 

'9 

9 

.64673 

26 

9.69425 

32 

o.3o  575 

9.95  248 

6 

4i 

20 

9 

.64  698 

25 

9.69  457 

32 

o.3o  543 

9.95  242 

40 

21 

9 

.64  724 

9.69  488 

31 

o.3o  5i2 

9.95  236 

39 

22 

9 

.64  749 

9.69  52O 

o.3o48o 

9.95  229 

38 

23 

9 

•64775 

9.69  552 

32 

o.3o448 

9.95  223 

37 

25 

32 

h 

24 

9 

.64  800 

26 

9.69  584 

o.3o4i6 

9.95  2I7 

36 

25 

9 

.64826 

9.69  6i5 

o.3o385 

9.95  211 

35 

26 

9 

.6485i 

25 

26 

9.69  647 

32 

o.3o  353 

9.95  2O4 

7 
f, 

34 

27 

9 

.64877 

9.69679 

o.3o  32i 

9.95  198 

6 

33 

28 

9 

.64  902 

9.69  710 

o.3o  290 

9.95   192 

32 

29 

9 

.64  927 

25 

9.69  742 

32 

o.3o  258 

9.95  i85 

7 

3i 

30 

9 

.64953 

9.69  774 

32 

o.3o  226 

9.95  179 

30 

L.  Cos. 

d. 

L.  Cotg.      d. 

L.  Tang. 

L. 

Sin. 

d. 

' 

63°  3D'. 

PP 

32 

31 

26 

25 

7 

6 

i 

3-2 

i'1 

! 

2.6 

2-5 

.! 

0.7 

0.6 

2 

6.4 

6.2 

2 

5-2 

5-° 

2 

1.4 

1.2 

3 

9.6 

9-3 

3 

7.8 

7-5 

3 

2.1 

1.8 

4 

12.8 

12.4 

4 

10.4 

10.  0 

4 

2.8 

2.4 

5 

16.0 

5 

13-0 

12.5 

3.5 

3-° 

6 

19.2 

18.6 

.6 

15-6 

15.0 

6 

4.2 

3-6 

7 

22.4 

21.7 

•7 

18.2 

17-5 

•7 

4-9 

4.2 

8 

'5-6 

24.8 

.8 

20.8 

20.0 

.8 

5-6 

4.8 

9 

23.4              22.5 

6.3              5-4 

82 


26°  3O . 


L.  Sin.       d. 

L.  Tang. 

d. 

L. 

Cotg. 

L.  Cos. 

d. 

30 

9.64  953 

9.69 

774 

o. 

3o  226 

9.95 

179 

g 

30 

3i 

9.64  978 
9.65  oo3 

25 

9.69 
9.69 

8o5 
837 

31 
32 

o. 

0. 

3o  195 
3o  i63 

9.95 
9.95 

178 

167 

6 

29 

28 

33 

9-65  029 

9.69 

868 

3i 

o. 

3o  1  32 

9.95 

1  60 

7 

27 

34 

9.65o54 

25 

9.69 

32 
9°°      « 

o. 

3o  100 

9.95 

1  54 

f> 

26 

35 

9.65  079 

9.69 

932 

0. 

3o  068 

9.95 

i48 

25 

36 

9-65  io4 

25 
26 

9.69 

963 

31 
32 

o. 

3oo37 

9.95 

i4i 

6 

24 

37 

9.65  i3o 

25 

9.699^5 

0. 

3o  oo5 

9.95 

1  35 

6 

23 

38 

9.65i55 

9.70  026 

o. 

29974 

9.95 

I29 

22 

39 

9.65  i8o» 

25 

9.70  o58 

32 

0. 

29  942 

9.95 

122 

7 
g 

2* 

40 

9-65  2o5 

25 

9.70  089 

31 

o. 

29  91  1 

9.95 

116 

g 

20 

4i 

42 

9.65  23o 
9.  65  255 

25 
25 

9.70 
9.70 

121 

i5a 

32 
31 

o. 
o. 

29  879 
29  848 

9.95 
9.95 

I  IO 

io3 

7 

19 

18 

43 

9-65  281 

9.70 

1  84 

32 

o. 

29  816 

9.95 

°97 

17 

44 

9.653o6 

25 

9.70 

2l5 

o. 

29785 

9.95 

o9o 

7 

6 

16 

45 

9.65  33i 

9.70247 

o. 

29  753 

9.95 

084 

i5 

46 

9-65  356 

25 

9.70 

278 

3* 

0. 

29  722 

9.95 

078 

i4 

47 

9.65  38i 

25 

9.70 

3o9 

0. 

29691 

9.95 

071 

6 

i3 

48 

9.  65  4o6 

9.70 

34i 

o. 

29  659 

9.95 

065 

12 

49 

9.6543i 

25 

9.70 

372 

31 

0. 

29  628 

9.95 

o59 

I  I 

50 

9.  65  456 

9.70  4o4 

0. 

29  596 

9.95 

052 

10 

5i 

9.6548i 

25 

9.70 

435 

31 

0. 

29  565 

9.95 

o46 

7 

9 

52 

9.65  5o6 

9.  70  466 

o. 

29  534 

9.95 

o39 

g 

8 

53 

9.  6553i 

25 

9.70498 

32 

o. 

29  5o2 

9.95 

o33 

7 

54 

9.65  556 

25 

9.70 

529 

o. 

29471 

9.95 

027 

7 

6 

55 

9.65  58o 

9.70 

56o 

o. 

29440 

9.95 

020 

g 

5 

56 

9.65  6o5 

25 

9.70 

592 

32 

0. 

29  4o8 

9.95 

oi4 

4 

57 

9.65  63o 

25 

9.70  623 

o. 

29  377 

9.95 

007 

7 

6 

3 

58 

9.65  655 

9.70  654 

0. 

29  346 

9.95 

OOI 

2 

59 

9.65  680 

25 

9.70  685 

31 

0. 

29315 

9.94 

995 

I 

60 

9.65  705 

25 

9.70  717 

32 

o. 

29  283 

9.94988 

0 

L.  Cos. 

d. 

L.  Cotg.      d. 

L. 

Tang. 

L.  Sin. 

d. 

f 

63°. 

PP 

32 

31 

26 

25 

24 

7           6 

x 

3.2 

3-1 

2.6 

.1        2.5 

2.4 

.1 

o.  7            o.  6 

.  2 

6.4 

6.2 

5-2 

2                  5-0 

4.8 

.2 

1.4                     1.2 

•3 

9.6 

9-3 

7.8 

•3            7-5 

7-2 

•3 

2.1                   1.8 

•4 

12.8 

12.4 

10.4 

.4          10.0 

9.6 

•4 

2.8            2.4 

.5 

16.0 

13.0 

r                  12.  5 

12.0 

5 

3-5            3-o 

.6 

19.2 

18.6 

15.6 

.6          15.0 

14.4 

.6 

4.2            3.6 

-? 

22.4 

21.7 

18.2 

•7          17-5 

16.8 

•7 

4-9            4-2 

.8 

25-6 

24.8 

20.8 

8              20.0 

19.2 

.8 

5.6            4.8 

•9 

27.9         23.4 

6-3             5-4 

83 


27°. 


/ 

L.  Sin. 

d. 

L.  Tang. 

d. 

L 

.  Cotg. 

L.  Cos. 

d. 

0 

9.66  705 

9.70717 

o 

29283 

9.94 

988 

60 

I 

9.65  729 

25 

9.70  748 

31 

o 

29  252 

9.94 

982 

59 

2 

9.65754 

9.70779 

o 

29  221 

9.94 

Q75 

58 

3 

9.66779 

9.70  810 

31 

o 

29  190 

9.94 

969 

57 

25 

31 

7 

4 

9.65  8o4 

24 

9.70  84i 

32 

o 

29  1  59 

9.94 

962 

6 

56 

5 

9.66828 

9.70873 

o 

29  127 

9.94 

g56 

55 

6 

9.65853 

25 

9.70904 

31 

o 

29  096 

9.94949 

7 

54 

25 

31 

6 

7 

9.60878 

24 

9.70935 

31 

o 

29065 

9.94943 

53 

8 

9.65  902 

9.70  966 

o 

29  o34 

9.94  936 

52 

9 

9.65  927 

9.70997 

31 

o 

29  oo3 

9.94930 

5i 

10 

9.65  962 

24 

9.71 

028 

31 

o 

28  972 

9-94 

92  3 

g 

50 

1  1 

9.65  976 

25 

9.71 

o5g 

31 

o 

28941 

9.94917 

6 

49 

12 

9.66  ooi 

9.71 

090 

o 

28  910 

9.94  911 

48 

i3 

9.66026 

25 

9.71 

121 

32 

o. 

28  879 

9.94904 

7 

6 

47 

i4 

9.66  o5o 

25 

9.71 

i53 

31 

o. 

28847 

9.94 

898 

46 

i5 

9.66075 

9.71 

1  84 

o, 

28816 

9.94 

891 

45 

16 

9.66  099 

9.71 

215 

o. 

28  785 

9.94 

885 

44 

25 

3» 

7 

J7 

9.66  124 

24 

9.71 

246 

31 

o. 

28754 

9-94 

878 

43 

18 

9.66  i48 

9.71 

277 

o. 

28  723 

9-94 

871 

42 

'9 

9.66  173 

9.71 

3o8 

o. 

28  692 

9.94 

865 

4i 

20 

9.66  197 

9.71 

339 

o. 

28661 

9.94  858 

40 

21 

9.66  221 

25 

9.71 

37o 

31 

o. 

2863o 

9.94852 

39 

22 

9.66  246 

9.71 

4oi 

0. 

28  599 

9.94 

845 

38 

23 

9.66  270 

9.71 

43i 

0. 

28569 

9.94 

839 

37 

25 

31 

7 

24 

9.66  295 

24 

9.71 

462 

31 

o. 

28538 

9-94 

832 

6 

36 

25 

9.66  319 

9.71 

49^ 

o. 

28  507 

9.94 

826 

35 

26 

9.66343 

24 

9.71 

524 

o. 

28476 

9.94 

819 

7 

34 

25 

31 

6 

27 

9.66368 

9.71 

555 

31 

o. 

28445 

9.94 

8i3 

33 

28 

9.66  392 

9.71 

686 

o. 

284i4 

9-94 

806 

32 

29 

9.66416 

24 

9.71 

617 

31 

o. 

28  383 

9.94 

799 

7 

3i 

30 

9.66  44i 

9.71 

648 

o. 

28352 

9.94 

793 

30 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L. 

Tang. 

L.  Sin.    !  d. 

62°  3O  . 

PP 

32 

31 

30 

25 

24 

7              6 

.! 

3-* 

3-i 

3-o 

•i        2.5 

2.4 

.x 

o.  7            o.  6 

.2 

6.4 

6.2 

6.0 

.2                 5.0 

4.8 

2 

1-4                   1.2 

•3 

9.6 

9-3 

9.0 

•3            7-5 

7.2 

3 

2.1                    1.8 

•4 

12.8 

12.4 

12.0 

.4         xo.o 

9.6 

4 

2.8            2.4 

•5 

16.0 

iS-5 

15.0 

•5          12.5 

12.0 

3-5            3-o 

.6 

19.2 

18.6 

18.0 

.6          15.0 

14.4 

6 

4.2            3.6 

•7 

22.4 

21.7 

21.  0 

•7          17-5 

16.8 

•7 

4.9            4.2 

.8 

25-6 

24.8 

24.0 

.8              20.0 

19.2 

.8 

5-6            4-8 

•9 

27.0         27.0 

.9                22.  S 

2r.6 

6.3              5-4 

84 


27°  30 


' 

L.  Sin. 

d. 

L.  Tang. 

d. 

L. 

Cotg. 

L.  Cos. 

d. 

30 

9.66441 

24 
24 
24 
24 
25 
24 
24 
24 
24 
24 

24 
25 
24 
24 
24 
24 
24 
24 
24 
23 

24 
24 
24 
24 
24 
24 
24 
23 
24 
24 

9.71 

648 

31 

30 
31 
31 
31 

31 
30 
31 
31 
30 

31 
31 

3* 
3° 

3° 
31 
30 

3° 
3° 
30 
30 
3° 

0.28  352 

9.94 

793 

7 

6 

7 
6 
7 
7 
6 
7 
6 
7 
7 

7 
7 

t 

3 

d 
3 

(. 

3 

(. 

3 
3 

e 

3 
3 
'i 

30 

3i 

32 

33 

34 
35 
36 

37 
38 
39 

9.66465 
9.66489 
9.665i3 

9.66537 
9.66  562 
9.66586 

9.66  610 
9.66634 
9.66658 

\O  vo  "O  VO  vO  VO  vo  VO  vO 

679 
709 
740 

771 
802 
833 

863 
894 
925 

0.28  32i 
0.28  291 
0.28  260 

0.28  229 
0.28  198 
0.28  167 

0.28  137 
0.28  106 
0.28  075 

9.94  786 
9.94  780 
9.94773 

9.94767 
9.94  760 
9.94753 

9.94747 
9.94  740 
9.94734 

29 

28 
27 
26 

25 

24 

23 
22 
21 

40 

9.66682 

9-71 

955 

o. 

28045 

9.94 

727 

20 

4i 

42 

43 

44 
45 
46 

47 

48 

49 

9.66  706 
9.6673i 
9.66  755 

9.66779 
9.66  8o3 
9.66  827 

9.6685r 
9.66  875 
9.66  899 

9-71 
9-72 
9-72 

9.72 
9.72 
9.72 

9.72 
9.72 
9-72 

986 
017 

o48 

078 
109 
i4o 

170 

2OI 

23l 

0. 

o. 
o. 

o. 
o. 

0. 

o. 
o. 

0. 

28014 
27983 

27  952 
27  922 

27891 
27  860 

27830 

27799 
27769 

9.94  720 
9.94  714 
9-94707 

9.94  700 
9.94694 
9.94687 

9.94680 
9.94674 
9.94  667 

'9 
1  8 

17 

16 
i5 
i4 

i3 

12 
I  I 

50 

9.66  922 

9-72 

262 

o. 

27738 

9.94 

660 

10 

5i 

52 

53 

54 
55 
56 

57 
58 

59 

9.66  946 
9.66  970 
9.66994 

9.67  018 
9.67  042 
9.67  066 

9.67  090 
9.67  1  13 
9.67i37 

9.72  293 
9.72  323 
9.72  354 

9.  72  384 
9.72  415 
9.72  445 

9.72476 
9.72  5o6 
9.72  537 

o. 

0. 

o. 

o. 
o. 
o. 

o. 
o. 
o. 

27  707 
27677 

27  646 

27  616 
27585 
27555 

27  524 
27494 
27463 

9-94654 
9.94647 
9.94640 

9.94  634 
9.94  627 
9.94  620 

9.94614 
9.94607 
9.94  600 

9 

8 

7 

6 
5 
4 
3 

2 

I 

60 

9.6-7  161 

9.72 

567 

o. 

27433 

9.94 

593 

0 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L. 

Tang. 

L.  Sin. 

d. 

' 

62°. 

PP 

.1 

2 

3 
4 

31 

30 

35 

34 

23 

7 

6 

3-1 

6.2 

9-3 
12.4 
i8!6 

21.7 
24.8 

0.6 

1.2 

1.8 

2-4 

3-° 
3-6 

4-2 

4.8 

5-4 

6.0 
9.0 

12.0 

I5.0 

18.0 

21.0 
24.0 

27.0 

5-o 
7-5 

10.  0 

12.5 
15-0 

17-5 

20.0 

22.  =; 

.3               4.8 

•3            7-2 

.4         9.6 

•  5              I2.O 

.6          14.4 

.7          16.8 
.8         19.2 

4-6 
6.9 

9-2 
"•5 
13-8 

16.1 
18.4 

.2 

•3 
•4 

1.4 

2.1 

2.8 

3-5 
4.2 

4-9 
5-6 

6-3 

85 


28°. 


' 

L.Sin. 

d. 

L.  Tang 

.     d. 

L.  Cotg. 

L.  Cos.      d. 

0 

9 

67  161 

9.72  567 

3i 
30 
3i 
30 
3i 
30 
30 
3i 
3° 
3i 

30 
30 
3i 
30 
3° 
3i 
30 
3° 
30 
3i 

30 
3° 
30 
30 
3i 
3° 
3° 
3° 
3° 
3° 

0.27  433 

9.94  593 

6 

60 

I 

2 

3 

4 
5 
6 

7 

8 

9 

9.67  185 
9.67  208 

9.67  232 

9.67  256 
9.67  280 
9.67  3o3 

9.67327 
9.67  35o 
9-67  374 

23 
24 
24 
24 
23 
24 
23 
24 

9.72  598 
9.72  628 
9.72  65g 

9.72  689 
9.72  720 
9.72  750 

9.72  780 
9.72  811 
9.72  84i 

0.27  402 

0.27  372 
0.27  34i 

0.27  3n 
0.27  280 
0.27  250 

O.27  220 
O.27   189 

0.27  iSg 

9-94  587 
9.94  58o 
9.94573 

9.94  567 
9.94  56o 
9.94553 

9.94  546 
9.94  54o 
9.94533 

7 
7 
6 

7 
7 
7 
6 

7 

7 

7 
6 
7 
7 
7 
7 
6 
7 
7 
7 

7 

6 
7 
7 
7 
7 
7 
6 
7 
7 

59 
58 
57 
56 
55 
54 

53 

52 

5i 

10 

9 

67  398 

9.72  872 

0.27  128 

9.94  526 

50 

1  1 

12 

i3 

i4 
i5 
16 

*7 

18 

[9 

9.67  4ai 
9.67  445 
9.67468 

9.67  492 
9.675i5 
9.67539 

9.67  562 
9.67  586 
9.67  609 

24 
23 
24 

23 
24 

23 
24 

23 

9.72  902 
9.72  932 
9.72  963 

9.72  993 

9.73  023 

9.73054 

9.73  o84 

9.73  n4 
9.73  i44 

0.27  098 
0.27  068 
0.27  037 

0.27  007 
0.26  977 
0.26  946 

0.26  916 
0.26  886 
0.26  856 

9.94  5  19 
9.94  5i3 
9.94  5o6 

9.94499 
9.94  492 
9.94485 

9.94479 
9.94  472 
9.94  465 

49 

48 

47 
46 
45 
44 

43 

42 

4i 

20 

9 

.67  633 

9.73  175 

0.26  825 

9-94458 

40 

21 
22 
23 

24 
25 

26 

27 
28 
29 

9 
9 

9 

9 
9 
9 

9 
9 
9 

.67656 
.67  680 
.677o3 

.67  726 
.67  750 
.67773 

.67  796 
.67  820 
.67843 

24 
23 
23 
24 
23 
23 
24 
23 

9.73  205 
9.73  235 
9.73  265 

9.73  295 
9.73  326 
9.  73356 

9.73  386 
9.73416 
9.73446 

0.26  795 
0.26  765 
0.26735 

0.26  705 
0.26  674 
0.26  644 

0.26  6i4 
0.26  584 
0.26  554 

9.94  45  1 

9.94445 
9.  94438 

9-94431 
9.94  424 
9.94417 

9.94  4  10 
9.94  4o4 
9.94  397 

39 
38 

37 

36 
35 
34 
33 

32 

3i 

30 

9 

.67866 

9.73476 

o.  26  524 

9.94  390 

30 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L. 

Sin.     d. 

' 

61°  30  . 

PP 

.1 

.2 

•3 

•4 
•5 
.6 

3 

9 

3* 

30 

24 

23 

.1 

.2 

•3 
•4 

:i 

•9 

7 

6 

11 

9-3 

12.4 

I5-S 
18.6 

21.7 
24.8 

27.9 

3-° 
6.0 
9.0 

12.0 
15.0 

18.0 

21.  0 
24.0 

.2 

-3 
•4 

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•7 
.8 

•9 

:i 

7-2 

9.6 

I2.O 
14.4 

16.8 
19.2 

a 

6.9 
9.2 
i3.8 

16.1 
18.4 

0.7 
1.4 

2.  I 

2.8 

3-5 

4-2 

4-9 
5-6 
6.3 

0.6 

1.2 

1.8 

2-4 

30 

3.6 

4.2 

4.8 

5-4 

86 


28°  3O 


L.  Sin.       d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

30 

9.67866 

9.73476 

3» 
30 
30 

3° 
3° 
3° 
30 
3<> 
30 
30 

3° 
30 
3° 
3° 
30 
30 
30 
3° 
3° 
3° 

30 
30 
29 
30 
3° 
3° 
3° 
3«> 
29 
3° 

0.26 

524 

9.94 

39o 

7 

7 
7 
7 
7 
6 

7 

7 
7 
7 

7 
7 
7 
7 
7 
7 
6 

7 
7 
7 

7 
7 
7 
7 
7 
7 
7 
7 
7 
7 

30 

3i 

32 

33 

34 
35 
36 

3? 
38 
39 

9.67  890 
9.67913 
9.67  936 

9.67959 
9.67  982 
9.68  006 

9.68  029 
9.68  o52 

9.68075 

23 
23 
23 
23 
24 
23 
23 
23 

9.735o7 
9.73537 
9.73567 

9-73597 
9.73627 
9.73657 

9.73  687 
9.73  717 
9.73  747 

0.26  493 
0.26  463 
0.26433 

o.264o3 
0.26  373 
0.26343 

0.26  3i3 
0.26283 
0.26  253 

9.94383 
9.94376 
9.94  369 

9.94  362 
9.94355 
9  .  94  349 

9.94342 
9.94335 
9.94  328 

29 

28 
27 

26 

25 

24 

23 
22 
21 

40 

9.68098 

23 

9.73777 

0.26 

223 

9.94 

321 

20 

4i 

42 

43 

44 
45 
46 

4? 
48 

49 

9.68   121 

9.68  i44 
9.68  167 

9.68  190 
9.68  21  3 

9.68  237 

9.68  260 

9.68283 

9.68  3o5 

23 

23 
23 
23 
23 
24 
23 
23 

22 

9.73807 
9.73837 
9.73  867 

9-73897 
9.73927 
9.73957 

9.73987 
9-74oi7 
9.74  047 

0.26  193 
0.26  i63 
0.26  i33 

0.26  io3 
0.26073 
0.26  o43 

0.  26013 

0.25983 

0.25  953 

9.94  3i4 
9.94  3o7 
9.94  3oo 

9.94  293 
9.94  286 
9.94  279 

9.94  273 
9.94  266 
9.94  259 

*9 

18 

'7 
16 
i5 
i4 

i3 

12 
II 

50 

9.68  328 

23 

23 
23 
23 
23 
23 
23 
23 
23 
22 
23 

9.74077 

0.25 

923 

9.94 

252 

10 

5i 

52 

53 

54 
55 
56 

5? 
58 
59 

9.68  35i 
9.68374 
9.68397 

9.68  420 
9.68443 
9.  68  466 

9.68489 
9.68  5i2 
9.68  534 

9.74  107 
9.74  i37 
9.74  166 

9.74  196 
9.74  226 
9.74  256 

9.74  286 
9.74  3i6 
9-74345 

0.26  893 
0.25863 
0.25834 

o.25  8o4 
o.25  774 
o.25  744 

o.25  714 
o.25  684 
0.25655 

9.94 
9.94 
9.94 

9.94 
9-94 
9.94 

9.94 
9-94 
9,94 

245 

238 

23l 

224 
2I7 
2IO 

203 

i96 
i89 

9 

8 

7 
6 
5 
4 
3 

2 
I 

60 

9.68  557 

9.74375 

0.25 

625 

9.94 

182 

0 

L.  Cos. 

d. 

L. 

Cotg. 

d.     L.  Tang. 

L.  Sin.      d. 

' 

61°. 

PP 

.2 

•3 
•4 

:! 

•9 

31 

30           29 

.1 

.2 

•3 

•4 

:! 
:i 

•9 

24 

33 

22 

7 

6 

1:1 

9-3 
12.4 

;i:i 

21.7 
24.8 

27.4 

3-o           2.9 
o.o          5.8 
9.0          8.7 

12.0             II.  6 

15.0      14.5 

18.0         17.4 

21.  0            20.3 

24.0        23.2 
27.0          26.1 

2.4 

4.8 

7-2 
9.6 

12.0 
14.4 

16.8 
19.2 

::i 

6.9 

9.2 
11.5 

13.8 

16.1 
18.4 

20.7 

2.2 

4-4 
6.6 

8.8 

II.  0 

13-2 

5* 

.2 

•3 
•4 

:I 
:I 

__ 

0.7 
1.4 

2.1 
2.8 

3-5 
4.2 

4.9 
5-6 
6-3 

0.6 

1.2 

1.8 

2.4 

rs 

ti 

5-4 

87 


- 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L. 

Cos.     d. 

0 

9 

.68  557 

9.74375 

3° 
3° 
3° 
29 

3° 
3° 
29 

3° 
3° 
30 
29 
3<> 
30 
29 
30 
30 
29 

30 
•29 
30 
29 
30 
3° 
29 

3° 
29 
3° 
29 
3° 
29 

o.25  625 

9.94  182 

7 
7 
7 
7 
7 
7 
7 
7 
7 
7 

7 
7 
8 

7 

7 
7 
7 
7 
7 

60 

I 

2 

3 

4 
5 
6 

8 
9 

9 

9 
9 

9 
9 
9 

9 

9 
9 

.68  58o 
.686o3 
.68625 

.68648 
.68671 
.68694 

.68  716 
.68  739 
.68  762 

23 

22 

23 
23 
23 
22 

23 
23 

9.74405 
9.74435 
9.74465 

9.74  494 
9.74  524 
9-74554 

9.  74  583 
9.74  6  1  3 
9-74643 

o.25  5g5 
o.25  565 
0.26  535 

o.25  5o6 
o.25  476 
o.25  446 

o.25  417 
o.25  387 

0.25  357 

9.94  175 

9.94  1  68 
9.94  161 

9.94  1  54 
9.94  147 
9.94  i4o 

9.94  i33 
9.94  126 
9.94  119 

59 
58 
57 
56 
55 
54 
53 

52 

5i 

10 

9 

.68  784 

9.74673 

0.25  327 

9.94  112 

50 

1  1 

12 

i3 

i4 
i5 
16 

17 

id 

'9 

9 
9 
9 

9 
9 
9 

9 
9 
9 

.68  807 
.68  829 
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.68875 
.68  897 
.68  920 

.68  942 
.68  965 

.68  987 

22 
23 
23 
22 
23 
22 

23 
22 

9.74  702 
9.74732 
9.74  762 

9.74  791 
9.74  821 
9.7485i 

9.74880 
9.74  910 
9.74939 

o.25  298 

0.25  268 

o.25  238 

o.25  209 
o.25  179 

0.2D  l49 
O.25  I2O 

o.25  090 
o.25  061 

9.94  105 
9.94  098 
9.94  090 

9.94083 
9.94  076 
9.94  069 

9.94  062 
9.94  o55 
9.94  o48 

49 
48 

47 

46 

45 
44 

43 

42 

4i 

20 

9 

.69  oio 

9.74969 

o.25  o3i 

9.94  o4i 

7 

40 

21 
22 
23 

24 
25 

26 

27 
28 
29 

9.69  o32 
9.69  o55 
9.69077 

9.69  100 

9.69  122 

9.69  i44 

9.69  167 
9.69  189 
9.69212 

23 
22 

23 
22 
22 

23 
22 

23 

9.74998 
9.75  028 
9.75  o58 

9.76087 
9.75  117 
9.75  i46 

9.75  176 
9.75  2o5 

9.75235 

O.25  OO2 

0.24  972 

0.24  942 

0.24  9i3 
0.24883 
0.24854 

0.24824 
0.24795 

0.24  765 

9.94  o34 
9.94  027 
9.94  020 

9.94  OI2 

9.94  oo5 

9.93998 
9.93991 

9.93  984 
9.93  977 

7 
7 
8 
7 
7 
7 
7 
7 
7 

39 

38 

37 

36 
35 

34 

33 

32 

3i 

30 

9 

.69234 

9.75  264 

0.24  736 

9-93  970 

30 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L. 

Sin. 

d. 

' 

60°  30  . 

PP 

.2 

•3 
•  4 

:i 

•Q 

30 

39 

23 

22 

.1 

.2 

•  3 
•4 

:I 

J 

8 

7 

3-° 
6.0 
9.0 

12.0 
15-0 

18.0 

21.0 
24.0 

11 

8.7 

ii.  6 

i4-5 
17.4 

20.3 
23-2 

.  i 

.2 

•3 

•4 

•5 
.6 

•7 
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6.9 

9-2 
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13.8 

16.1 
18.4 
20.7 

2.2 

4-4 
6.6 

8.8 

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13-2 

15-4 

17.6 

iq.  8 

0.8 
1.6 
2.4 

3-2 

4.0 
4.8 

5-6 
6.4 

0.7 
1.4 

2.1 
2.8 

3-5 
4-2 

4.9 
*' 

88 


29°  30 . 


' 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

30 

9 

69  234 

22 

23 
22 
22 
22 
23 
22 
22 
22 

9.76  264 

30 
29 

3° 
29 
29 

3° 
29 
3° 
29 
29 

3° 
29 
3° 
29 
29 
30 
29 
29 
29 
3° 
29 
29 
29 
3° 
29 
29 

29 
30 
29 
29 

0.24  736 

9.93970 

30 

3i 

32 

33 

34 
35 
36 

37 
38 
39 

9 
9 
9 

9 
9 
9 

9 
9 
9 

69  256 
69  279 
69  3oi 

69  323 
69  345 
69  368 

69  Sgo 
69  412 
69434 

9.75  294 

9.75  323 
9.75  353 

9.75382 
9.75  4n 
9.75441 

9.75470 
9.75  500 
9.75  529 

0.24  706 
0.24  677 
0.24  647 

0.24  618 
0.24  589 
0.24  559 

0.24  53o 
0.24  5oo 
0.24  471 

9.93  963 
9.93955 
9.93  948 

9.93  941 
9.93  934 
9.93927 

9.93  920 
9.939i2 
9.93  9o5 

8 
7 
7 
7 
7 
7 
8 

7 

7 

7 
7 
8 

7 
7 
7 
8 
7 
7 
7 

7 

8 

7 
7 
8 
7 
7 
7 
8 
7 

29 

28 
27 

26 

25 
24 

23 
22 
21 

40 

9 

69456 

9.75558 

0.24  442 

9.93  898 

20 

4i 

42 

43 

44 
45 
46 

47 
48 

49 

9 
9 
9 

9 
9 
9 

9 
9 
9 

69479 
69  5oi 
69  523 

69  545 
69  567 
69  589 

69  61  1 
69  633 
69  655 

22 
22 
22 
22 
22 
22 
22 
22 

9.75  588 
9.75617 
9.75  647 

9.75676 
9.75  705 
9-75735 

9.75  764 
9<75  793 
9.75  822 

0.24  4i2 
0.24383 
0.24353 

0.24  324 
0.24  295 
0.24  265 

0.24  236 
0.24  207 
0.24  178 

9.93  891 
9.93  884 
9.93  876 

9.93  869 
9.93  862 
9.  93855 

9-93847 
9.93  84o 
9.93  833 

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18 

17 
16 
i5 
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12 
II 

50 

9 

69677 

9.75  852 

0.24  i48 

9.93  826 

10 

5i 

52 

53 

54 
55 
56 

57 

58 

59 

9.69699 
9.69721 
9.69  743 

9.69  765 
9.69787 
9.69  809 

9.69  83i 
9.69853 
9.69875 

22 
22 
22 
22 
22 
22 
22 
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9.75  881 
9.75  910 
9.75  939 

9.75969 
9.75998 
9.76027 

9.76  o56 
9.  76  086 
9.76  115 

0.24  119 
0.24  090 
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0.24  o3i 
0.24  002 
o.23  973 

0.23  944 

o.23  914 
0.23885 

9.93  819 
9.93  81  1 
9.93  8o4 

9.93  797 
9.93  789 
9.93  782 

9-93775 
9.93  768 
9.93  760 

9 

8 

7 
6 
5 
4 
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2 
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9.76  i44 

0.23856 

9.93  753 

0 

L.  Cos.      d. 

L.  Cotg. 

d. 

L.  Tang. 

L. 

Sin.     d. 

' 

6O°. 

PP 

•  2 

•3 

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30 

29 

23 

22 

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3-° 
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9.0 

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18.0 

21.0 
24.0 

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8.7 

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M-5 
17-4 

20.3 

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•3 
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6.9 

9.2 

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16.1 
18.4 

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2.2 

4-4 
6.6 

8.8 

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13.2 

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17.6 
19.8 

0.8 
1.6 

a-4 

3-2 

4'R 
4.8 

5-6 
6-4 

7.2 

0.7 
i-4 

2,1 

2.8 

3-5 

4-2 

4-9 

I'6 

6-3 

89 


3O°. 


' 

L.Sin 

d. 

L.  Tang. 

d. 

L 

Cotg. 

L.Cos.    |d. 

0 

9.69  897 

9.76 

1  44 

o. 

23856 

9.93 

753 

60 

I 

9.69919 

22 

9.76 

178 

29 

o. 

23  827 

9.93 

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a 

59 

2 

9.69  941 

9.76 

202 

0. 

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9-93 

738 

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3 

9.69  963 

21 

9-76 

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30 

0. 

23769 

9.93 

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7 
7 

57 

4 

9.69  984 

22 

9-76 

261 

29 

o. 

23739 

9.93 

•724 

7 

56 

5 

9.70  006 

9.76 

29O 

0. 

23  7IO 

9.93 

7i7 

55 

6 

9.70  028 

9.76 

3i9 

o. 

2368i 

9.93 

7°9 

54 

22 

29 

7 

7 

9.  70  050 

9.76 

348 

29 

o. 

23652 

9.93 

702 

53 

8 

9.70  072 

9.76 

377 

o. 

23623 

9.93 

695 

52 

9 

9.7o  093 

9.76  4o6 

29 

o. 

23594 

9.93 

687 

5i 

10 

9.7o  n5 

9.76435 

0. 

23  565 

9.93 

680 

50 

ii 

9.7o  1  3 

7 

9-76464 

29 

o. 

23  536 

9.93 

673 

8 

49 

12 

9.70  i5g 

9.76 

493 

0. 

23  507 

9.93  665 

48 

i3 

9.70  180 

9.76 

522 

29 

o. 

23478 

9.93  658 

7 

47 

i4 

9.70  202 

22 

9.76 

55i 

29 

0. 

23449 

9.93  65o 

8 

46 

i5 

9  .70  224 

22 

9.76 

58o 

o. 

23  42O 

9.93  643 

45 

16 

9.70  245 

21 

9.76 

609 

29 

o. 

23  Sgi 

9.93636 

7 

44 

17 

9.70  267 

22 

9.76  639 

3° 

o. 

2336i 

9.93 

628 

43 

18 

9.70  288 

9.76 

668 

o. 

23332 

9.93 

621 

42 

X9 

9.70  3io 

22 

9.76697 

29 

0. 

23  3o3 

9-93 

6i4 

7 

g 

4i 

20 

9.70  332 

22 

9.76 

725 

0. 

23275 

9.93  606 

40 

21 

9.70  353 

9.76 

754 

29 

o. 

23  246 

9.93 

599 

7 

8 

39 

22 

9.70  375 

9-76 

783 

0. 

23  217 

9.93 

5oi 

38 

23 

9.70  396 

9-76 

812 

29 

o. 

23  188 

9-93 

584 

7 

37 

22 

29 

7 

24 

9.70  4i8 

9-76 

84  1 

29 

o. 

23  i5g 

9-93 

577 

8 

36 

25 

9.7o439 

9-76 

87o 

o. 

23  i3o 

9-93 

569 

35 

26 

9.7o  46i 

9.76899 

o. 

23  IOI 

9.93 

562 

7 

34 

21 

29 

K 

27 

9.70  482 

9<76  928 

29 

o. 

23  072 

9.93 

554 

33 

28 

9.7o  5o4 

9.76 

957 

o. 

23o43 

9.93 

547 

32 

29 

9.7o  525 

9.76  986 

29 

o. 

23  Ol4 

9.93 

539 

3i 

30 

9.7o  547 

.9-77 

015 

0. 

22  985 

9.93 

532 

7 

30 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L. 

Tang. 

L.  Sin. 

d. 

' 

59°  3O  . 

PP         30 

29 

28 

22 

21 

8               7 

•  i            3-0 

2.9 

2.8 

.1                2.2 

2.1 

.1 

0.  8                   0.  7 

.2                  6.0 

5-8 

5-6 

.2                 4.4 

4.2 

.2 

1.6            1.4 

•3            9-o 

8.7 

8.4 

.3           6.6 

6.3 

•  3 

2.4                    2.1 

•  4              12.0 

n.6 

II.  2 

.4           8.8 

8.4 

•4 

3.2            2.8 

•5         15-0 

14-5 

14.0 

•  5          ii-o 

10.5 

•5 

4-°            3-5 

.6         18.0 

17.4 

16.8 

.6          13.2 

12.6 

.6 

4.8            4.2 

.7         210 

20.3 

19.6 

•7          15-4 

14-7 

•7 

5.6            4.9 

.8         24.0 

23.2 

22.4 

.8          17.6 

16.8 

.8 

6.4            5.6 

•9          27.0 

.9          19.8 

18.9 

.9 

9° 


30°  30 


t 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

30 

3i 

32 

33 

34 
35 
36 

37 

38 

39 

9- 

70547 

9-77  015 

29 
29 
28 
29 
29 
29 
29 
29 
28 
29 

29 
29 
29 
28 
29 
29 
29 
28 
29 
29 
28 
29 
29 
29 
28 
29 
28 
29 
29 
28 

0.22  985 

9.93  532 

30 

9- 
9- 

9- 

9- 

9- 
9- 

9- 
9- 
9- 

70  568 
70  5go 
70  61  1 

70  633 
70654 
70  675 

70697 
70  718 
70  739 

22 
21 
22 
21 
21 
22 
21 
21 

9.77  o44 
9.77073 
9.77  101 

9.77  i3o 
9.77  i59 
9.77  1  88 

9.77217 
9.77  246 
9-77  274 

O.  22  956 
0.22  927 
O.22  899 

0.22  870 
O.22  84l 
O.22  8l2 

0.22  783 
0.22  754 
O.22  726 

9.93525 
9.93517 
9.93  5io 

9.93  5o2 

9.93495 
9.93487 

9.93480 
9.93  472 
9.93  465 

8 

7 
8 

7 
8 

7 
8 

7 

29 

28 
27 

26 

25 

24 

23 
22 
21 

40 

9- 

70  761 

9.773o3 

O.22  697 

9.93457 

7 
8 
7 
8 

7 
8 

7 
8 

7 
8 

7 
8 

7 
8 
8 
7 
8 

7 
8 

7 

20 

4i 

42 

43 

44 
45 
46 

47 
48 

49 

9- 
9- 
9 

9- 

9 
9 

9 

9 
9 

70  782 
70  8o3 
70824 

70  846 
70  867 
70  888 

70909 
7093i 
70  952 

21 
21 
22 
21 
21 
21 
22 
21 

9.  77  332 
9.77  36i 
9.77  390 

9.77418 
9.77447 
9.77476 

9.77  505 
9.77533 
9.77  562 

O.22  668 
O.22  639 
0.22  6lO 

O.22  582 

0.22  553 

0.22  524 

O.22  4g5 
0.22  467 
0.22  438 

9.93450 
9.93  442 
9.93435 

9.93427 
9.93  420 
9.93  412 

9.93405 
9.93  397 

9.93  390 

1  8 

16 
i5 
i4 
i3 

12 
II 

50 

9 

70973 

21 

21 
21 
22 
21 
21 
21 
21 
21 
21 

9-77  59i 

0.22  409 

9.93  382 

10 

5i 

52 

53 

54 
55 
56 

57 
58 
59 

9 
9 
9 

9 

9 

9 
9 
9 

70994 
71  oi5 
71  o36 

71  o58 
71  079 
71  100 

71    121 

71  142 
71  i63 

9-77  619 
9.77  648 
9.77  677 

9.77  706 
9.77  734 
9.77763 

9.77  791 
9.77820 
9.77  849 

O.22  38l 
0.22  352 
0.22  323 

O.22  294 
O.22  266 
0.22  237 

O.22  209 
O.22   l8o 
O.22   l5l 

9.93375 
9.93367 
9.93  36o 

9.93  352 
9.93344 
9.93337 

9.93329 

9.93  322 

9.93  3i4 

9 

8 

7 
6 
5 
4 

3 

2 

I 

60 

9 

.71  1  84 

9.77  877 

0.22   123 

9-93  3o7 

0 

L.  Cos. 

d. 

L.  Cotg.      d. 

L.  Tang. 

L. 

Sin. 

d. 

59°. 

PP 

.1 

2 

3 

4 

•  7 
.8 
•9 

39 

28 

22 

21 

.1 

2 

•3 
•4 

•7 
.8 

•9 

8                7 

2.9 

5-8 

8.7 

ii.  6 
"4-5 
17.4 

20.3 
23.2 
26.1 

2.8                            .1 

5-6 
8.4                  .3 

II.  2                            4 

14.0                  .5 
16.8                  .6 

19.6                  .7 

22.4 

25.2                    .9 

2.2 

4-4 
6.6 

8.8 

II.  O 

13-2 

\7.6 

19.8 

2.  I 
4-2 
6-3 

8.4 
10.5 

12.6 

14.7 
16.8 
18.9 

o.  8              o.  7 
1.6              1.4 

2  4                    2.1 

3.2              2.8 
4-o              3-5 
4.8              4.2 

5-6              4-9 
6.4               5.6 

31°. 


/ 

L.  Sin.      d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

0 

9.71  1  84 

9.77877 

O.22   123 

9.93  307 

60 

I 

.71  205 

9.77906 

29 

0.22  094 

9.93  299 

g 

5p 

2 

9.71  226 

9-77  935 

28 

0.22  o65 

9.93  291 

58 

3 

9.71  247 

9.77963 

0.22  037 

9.93  284 

' 

57 

29 

K 

4 

9.71  268 

21 

9.77992 

28 

0.22  008 

9.93  276 

56 

5 

9.71  289 

9.78  020 

O.2I  980 

9.93  269 

55 

6 

9.71  3io 

21 

9.78  049 

28 

O.2I  95  I 

9.93  261 

8 

54 

7 

9.71  33i 

21 

9.78077 

0.21  923 

9.93  253 

53 

8 

9.71  352 

9.78  106 

0.21  894 

9.93  246 

52 

9 

9.71  373 

9.78135 

29 
28 

0.21  865 

9.93  238 

5i 

10 

9 

.71  393 

9.78  i63 

20 

0.21  837 

9.93  23o 

50 

ii 

9 

.71  4i4 

21 

9.78  192 

28 

0.21   808 

9.93  223 

7 
g 

49 

12 

9 

.7i435 

9.78  220 

O.2I   780 

9.93  21  5 

48 

i3 

9 

.71  456 

21 

9.78  249 

28 

0.21  751 

9.93  207 

8 

47 

i4 

9 

.71  477 

21 

9.78277 

29 

O.2I   723 

9.93  2OO 

7 
g 

46 

i5 

9 

.71498 

9.78  3o6 

28 

0.21   694 

9.93  192 

45 

16 

9 

.71  5i9 

2O 

9.78334 

29 

0.21  666 

9.93  184 

44 

17 

9 

.71  539 

21 

9.78  363 

28 

0.21  637 

9.93  177 

g 

43 

18 

9 

.71  56o 

9.7839i 

?8 

0.21   609 

9.93  169 

42 

'9 

9 

.71  58i 

9.78419 

0.21  58i 

9.93  161 

4i 

20 

9 

.71  602 

9.78448 

28 

O.2I   552 

9.93  i54 

7 

40 

-  21 

9.71  622 

21 

9.78476 

29 

O.2I   524 

9.93  i46 

g 

39 

22 

9 

.71  643 

9-78505 

28 

O.2I  495 

9.93  i38 

38 

23 

9 

.71  664 

9.78533 

O.2I  467 

9.93  i3i 

7 

37 

21 

29 

X 

24 

9 

.71685 

9.78  562 

28 

0.21  438 

9.93  123 

g 

36 

25 

9 

.71  705 

9.78  590 

28 

0.21  4 

to 

9.93  n5 

35 

26 

9 

.71  726 

9.78  618 

0.21   382 

9.93  108 

7 

34 

21 

29 

H 

27 

9 

.71  747 

9.78  647 

28 

0.21  353 

9.93  100 

g 

33 

28 

9.71  767 

9.78675 

O.2I   325 

9  .  93  092 

32 

29 

9.71  788 

9.78  704 

28 

O.2I  296 

9.93084 

3r 

30 

9 

.71  809 

9.78732 

O.2I  268 

9.93077 

7 

30 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L. 

Sin. 

d. 

' 

58°  30  . 

PP 

29 

28 

21 

20 

8 

7 

, 

2.9 

2.8 

j 

2.1 

2.0 

.1 

0.8 

0.7 

2 

5.1 

5-6 

.2 

4.2 

4.0 

.2 

1.6 

1.4 

•3 

8-7 

8.4 

•  3 

6.3 

6.0 

•3 

2.4 

2.  1 

•4 

ii.  6 

II.  2 

•4 

8.4 

8.0 

•4 

3«2 

2.8 

•5 

*4*  5 

14.0 

•5 

10.5 

10.  0 

•  5 

4.0 

3-5 

.6 

17-4 

1  6.  8 

.6 

12.6 

12.0 

.6 

4.8 

4.2 

•7 

20.3 

19.6 

.7 

14.7 

14.0 

.7 

5-6 

4.9 

.8 

23.2 

22.4 

.8 

16.8 

16.0 

.8 

6.4 

5-6 

•9 

18.9            18.0 

•9 

7.2              6.3 

92 


31°  3O . 


, 

L.  Sin.       d. 

L.  Tang. 

d. 

L.  Cotg. 

L. 

Cos. 

d. 

30 

9 

.71  809 

9.78732 

28 
29 
28 
28 
29 
28 
28 
29 
28 
28 

28 
29 
28 
28 
28 
29 
28 
28 
28 
28 

29 
28 
28 
28 
28 
28 
29 
28 
28 
28 

O.2I   268 

9.93077 

8 
8 
8 

7 
8 
8 
8 
8 

7 

8 

8 
8 

7 
8 
8 
8 
8 
8 

7 
g 

30 

3i 

32 

33 

34 
35 
36 

37 
38 

39 

ooo  ooo  ooo 

.71  829 
.71850 
.71  870 

.71  891 
.71  9n 
.71932 

.71  952 
.71  973 

.71  994 

21 
20 
21 
2O 
21 
20 
21 
21 

9.78  760 
9.78789 
9.78817 

9.78845 
9.78874 
9.78  902 

9.78  930 
9.78959 
9.78987 

O.2I   24O 
O.2I   211 
0.21    183 

0.21  155 

0.21    126 
O.2I  098 

0.21   070 
O.2I  o4l 
0.21   Ol3 

9.93  069 
9.93  061 
9.93  o53 

9.93  o46 

9.93  o3o 

9.93  022 
9.93  oi4 
9.93  007 

29 
28 
27 
26 

25 
24 

23 
22 
21 

40 

9 

.72  014 

9.79015 

O.2O  985 

9.92999 

20 

4i 

42 

43 

44 
45 
46 

47 
48 

49 

ooo  ooo  ooo 

.72  o34 
.72055 
.72  o75 

.72  096 
.72  1  16 

.72l37 

.72  157 
.72  177 
.72  198 

21 
20 
21 
2O 
21 
20 
20 
21 

9.79  o43 
9.79072 
9.79  100 

9.79  128 

9.79  i56 
9.79185 

9.79  2i3 
9.79  241 

9.79269 

O.2O  957 
O.2O  928 
O.2O  900 

O.2O  872 
O.2O  844 
O.2O  8l5 

0.20  787 
O.2O  759 
O.2O  781 

9.92  991 
9.92  983 
9.92976 

9.92  968 
9.92  960 
9.92  952 

9.92  944 
9.92  936 
9.92929 

18 
17 

i5 
i4 
i3 

12 
II 

50 

9 

.72  218 

9.79297 

0.20  703 

9.92  921 

8 
8 
8 
8 
8 

7 
8 
8 
8 
g 

10 

9 

8 

7 

6 

5 

4 

3 

2 
I 

5i 

52 

53 

54 
55 
56 

57 
58 
59 

9 

9 

9 

9 
9 
9 

9 

9 
9 

.72  238 
.72  259 
.72  279 

.72  299 
.72  32o 
.72  34o 

.72  36o 
.72  38i 
.72  4oi 

21 

20 
20 
21 
20 
20 
21 
20 

9.79  326 

9.79354 
9.79382 

9.79410 
9.79438 

9.79  466 

9.79495 
9.79  523 
9.79  55i 

O.2O  674 

0.20  646 
0.20  618 

o  •  20  5oo 

0.20  562 

0.20  534 
0.20  5o5 

O.2O  477 
O.2O  449 

9.92  913 
9.92  905 
9.92897 

9.92  889 
9.92  881 
9.92  874 

9.92  866 
9.92  858 
9.92  850 

60 

9 

.72  421 

9-79  579 

0.2O  421 

9.92  842 

0 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L. 

Sin. 

d. 

t 

58°. 

PP 

.1 

.2 

•3 

•4 
•  5 
.6 

•9 

29 

28 

21 

20 

2 

3 
4 

1 

9 

8 

7 

ii.  6 
14.5 
17.4 

20.3 
23.2 

26.1 

2.8 

8.4 

II.  2 

14.0 

16.8 

19.6 
22.4 
25.2 

.1 

.2 

•  3 
•4 

J 

•  7 
.8 
•9 

2.1 
4-2 
6-3 

8.4 
10-5 
12.6 

14.7 

16.8 
18.9 

2.0 

4.0 

6.0 
8.0 

IO.O 
12.0 

14.0 
16.0 
18.0 

0.8 
1.6 
2.4 

3-2 

ts 

0.7 
1.4 

2.1 

2.8 

3-5 
4.2 

4.9 

y 

93 


32°. 


, 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos.     d. 

0 

9.72  421 

9.79  579 

28 

O.2O 

421 

9-92 

842 

D 

60 

I 

9.72441 

20 

9-79  6°7 

28 

0.20 

393 

9.92 

834 

8 

59 

2 

9.72  46i 

9.79635 

0.20 

365 

9.92 

826 

r> 

58 

3 

9.72  482 

2O 

9.79  663 

28 

0.2O 

337 

9.92 

818 

8 

57 

4 

9.72  602 

20 

9.79691 

O.2O 

3o9 

9.92 

810 

7 

56 

5 

9.72  522 

9.79  719 

28 

0.20  28l 

9.92 

8o3 

| 

55 

6 

9.72  542 

20 

9.79  747 

29 

O.2O 

253 

9.92 

795 

8 

54 

7 

9.72  562 

2O 

9.79776 

28 

O.2O 

224 

9.92 

787 

8 

53 

8 

9.72  582 

9.79  8o4 

O.2O 

196 

9.92 

779 

0 

52 

9 

9.72  602 

9.79  832 

0.20 

168 

9.92 

771 

8 

5i 

10 

9.72  622 

9.79  860 

O.2O  l4o 

9.92 

763 

0 

50 

1  1 

9.72  643 

2O 

9.79888 

28 

0.20 

112 

9.92 

7*5 

t 

49 

12 

9.72  663 

9.79916 

0.20  o84 

9.92 

747 

48 

i3 

9.72  683 

9.79944 

28 

O.2O 

o56 

9.92 

47 

20 

28 

i4 

9.72  703 

9.79972 

28 

O.2O 

028 

9.92 

73i 

I 

46 

i5 

9.72  723 

9.80  ooo 

O.2O 

ooo 

9.92 

723 

45 

16 

9.72  743 

9.80  028 

28 

0.19  972 

9.92 

7i5 

44 

17 

9.72763 

2O 

9.80  o56 

28 
28 

o.  19 

944 

9.92 

7°7 

I 

43 

18 

9.72783 

9.80084 

o.  19 

916 

9.92 

699 

f 

42 

'9 

9.72  8o3 

9.80  112 

28 

o.  19 

888 

9.92 

691 

5 

4i 

20 

9.72  823 

9.80  i4o 

o.  19 

860 

9.92 

683 

40 

21 

9.72843 

9.80  1  68 

o.  19 

832 

9.92 

675 

| 

39 

22 

9.72  863 

9.80  195 

o.  19 

805 

9.92 

667 

38 

23 

9.72  883 

19 

9.80  223 

28 

0.19 

777 

9.92 

659 

1 

37 

24 

9.72  902 

9.80  25i 

28 

o.  19 

749 

9.92 

65i 

| 

36 

25 

9.72  922 

9.80  279 

o.  19 

•721 

9.92 

643 

35 

26 

9.72  942 

20 

9.80  307 

28 

0.19  693 

9.92 

635 

1 

34 

27 

9.72  962 

9.  80  335 

28 

o.  19 

665 

9.92 

627 

| 

33 

28 

9.72  982 

9.  80  363 

o.  19 

637 

9.92 

6i9 

32 

29 

9.73  002 

20 

9.80  391 

28 

o.  19 

6o9 

9.92 

611 

3i 

30 

9.73  022 

9.80  419 

o.  19 

58i 

9.92 

6o3 

30 

L.  Cos. 

d. 

L. 

Cotg.      d. 

L.  Tang. 

L.  Sin. 

d. 

57°  30. 

PP 

29 

28             27 

21 

20 

19 

8 

7 

.1 

2.9 

2.8            2.7 

.1 

2.1 

2.0 

1.9 

.1 

0.8 

0.7 

.2 

5-8 

5-6            5-4 

.2 

4-2 

4.0 

3 

.2 

1.6 

1.4 

•3 

8.7 

8.4            8.1 

•3 

6.3 

6.0 

5-7 

•3 

2.4 

2.  I 

•4 

n.6 

II.  2               10.8 

•4 

8.4     '        8.0 

7.6 

•4 

3-2 

2.8 

•  5 

14.5 

14.0        13.5 

•5 

10.5           10.0 

9-5 

.  5 

4.0 

3-5 

.6 

i7-4 

16.8          16.2 

.6 

12.6 

12.0 

11.4 

;     -6 

4.8 

4-2 

•7 

20.3 

19.6          18.9 

•7 

14.7 

14.0 

»3*3 

!     .7 

5-6 

4-9 

.8 

22.4          21.6 

.8 

16.8 

16.0 

15.2 

.8 

6.4 

.9        26.1 

25.2          24.3 

18.9           18.0 

17.1 

•9 

7.2 

94 


32°  3D 


- 

L.  Sin.       d. 

L.  Tang. 

d.      L.  Cotg. 

L.  Cos. 

d. 

30 

9 

73  022 

20 

20 
2O 
20 

»9 
20 
2O 
20 

'9 

9.80419 

28 
27 
28 
28 
28 
28 
28 
28 
27 
28 

28 
28 
28 
27 
28 
28 
28 
27 
28 
28 

28 
27 
28 
28 
27 
28 
28 
27 
28 
28 

0.19  58i 

9.92  6o3 

30 

3i 

32 

33 

34 
35 
36 

37 
38 
39 

9 

9 
9 

9 
9 
9 

9 
9 
9 

73o4i 
73  061 
73  081 

73  101 

73  121 

73  i4o 

73  160 
73  180 
73  200 

9.80447 
9.80  4?4 
9.80  5o2 

9.8o53o 
9.  80  558 
9.80  586 

9.80  6i4 
9.80642 
9.80  669 

0.19553 
o.  19  526 
0.19498 

0.19470 
0.19  442 
o.  19  4i4 

0.19386 
0.19358 
0.19331 

9.92595 
9.92  587 
9-92579 
9.92571 
9.92  563 
9.92555 

9.92  546 
9.92  538 
9.92  53o 

8 
8 
8 
8 
8 

9 
8 
8 
g 

29 

28 
27 

26 

25 
24 

23 
22 
21 

40 

9 

73  219 

9.80  697 

0.19303 

9.92  522 

8 

8 
8 
8 
8 

9 
8 
8 
8 
8 

8 
8 

9 
8 
8 
8 

8 
8 

9 
8 

20 

4i 

42 

43 

44 
45 
46 

47 
48 

49 

ON  ON  ON  ON  ON  ON  ON  ON  ON 

73239 
73259 

73278 

73298 
73  3i8 
73337 

73357 
73377 
73396 

20 

'9 
20 
20 

19 

20 

20 

»9 

2O 

'9 

20 

20 

'9 
20 

2O 

19 
2O 

9.80725 
9.80  753 
9.80  781 

9.80808 
9.80  836 
9.80864 

9.80  892 
9.80  919 
9.80  947 

o.  19  275 
o.  19  247 
o.  19  219 

0.19  192 
0.19  1  64 
0.19  i  36 

0.19  108 
0.19081 
o.  19  o53 

9.92  5i4 
9.92  5o6 
9.92498 

9.92  490 
9.92  482 
9.92473 

9.92  465 
9.92457 
9.92  449 

18 

16 
i5 
i4 
i3 

12 
I  I 

50 

9 

734i6 

9.80975 

O.  19  O25 

9.92441 

10 

5i 

52 

53 

54 
55 
56 

57 
58 
59 

9 
9 
9 

9 
9 
9 

9 
9 
9 

73435 
73455 
73474 

73494 
735i3 
73533 

.73552 
.73572 
.73  591 

9.81  oo3 
9.81  o3o 
9.81  o58 

9.81  086 
9.81  ii3 
9.81  i4i 

9.81  169 
9.81  196 
9.81  224 

0.18997 
o.  1  8  970 
o.i  8  942 

0.18  9i4 
0.18  887 
0.18  859 

0.18  83i 
o.i  8  8o4 
o.  18  776 

9.92433 
9.92425 
9.92  4i6 

9.92  4o8 
9.92  4oo 
9.92  392 

9.92  384 
9.92  376 
9.92  367 

9 

8 

7 

6 
5 
4 
3 

2 
I 

60 

9 

.73611 

9.81  252 

o.i  8  748 

9.92  359 

0 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L. 

Sin. 

d. 

' 

57°. 

PP 

.2 

•3 

•4 
•5 
.6 

•9 

28 

27 

20 

19 

.2 

•3 
•4 

•7 
.8 

9                8 

2.8 

5-6 

8.4 

II   2 
I4.0 

16.8 

19.6 
22.4 

2.7            .1 

5-4                   -2 
8.1                   .3 

10.8                  .4 
ll'.l                 '.I 

18.9                  .7 
21.6                   .8 
24-3                   -9 

2.O 
4-0 

6.0 
8.0 

10.0 
12.  0 

14.0 

16.0 

18.0 

1.9 

3-8 

5-7 

7-6 
9-5 
11.4 

»5-2 
17.1 

o.  o              o.  8 
1.8               1.6 
2.7              2.4 

3.6              3.2 
4.5              4.0 

5-4              4-8 

6.3              5-6 
7.2              6.4 
8.  i              7.  2 

33°. 


' 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L. 

Cos. 

d. 

0 

9.73  611 

9.81  252 

27 

0.18  748 

9.92359 

60 

I 

9.73  63o 

20 

9.81   279 

28 

o.  18  721 

9.92  35i 

8 

59 

2 

9.73  650 

9.81  307 

28 

0.18  693 

9.92  343 

58 

3 

9.73  669 

20 

9.81  335 

27 

o.i8665 

9.92335 

9 

57 

4 

9.73  689 

19 

9.81  362 

28 

0.18  638 

9.92  326 

8 

56 

5 

9.73708 

9.81  390 

28 

o.  18  610 

9.92  3i8 

55 

6 

9.73  727 

I9 

9.81  4i8 

0.18  582 

9.92  3io 

54 

20 

27 

8 

7 

9 

.73747 

9.81  445 

28 

0.18  555 

9.92  3o2 

53 

8 

9 

.73766 

9  81  473 

0.18  527 

9.92  293 

52 

9 

9 

.73785 

19 

9.81  5oo 

27 

28 

o.  18  500 

9.92  285 

g 

5i 

10 

9 

.73805 

9.81  628 

28 

o.  18  472 

9.92  277 

g 

50 

1  1 

9 

.73824 

19 

9.8i  556 

27 

o.i  8  444 

9.92  269 

Q 

49 

12 

9 

.73  843 

9.8i  583 

0.18  417 

9.92  260 

48 

i3 

9 

.73863 

19 

9.8i  61 

I 

27 

0.18  389 

9.92  252 

8 

47 

i4 

9 

.73882 

19 

9.8i  638 

28 

0.18  362 

9.92  244 

Q 

46 

i5 

9 

.73901 

9.8i  666 

0.18  334 

9.92  235 

45 

16 

9 

.73921 

9.8i  693 

27 
28 

0.18  307 

9.92  227 

8 

44 

17 

9 

.7394o 

19 

9.81  721 

27 

o.  18  279 

9.92  219 

8 

43 

18 

9 

.73959 

9.81  748 

_Q 

O.  l8  252 

9.92  211 

42 

19 

9 

.73978 

9.81  776 

o.  18  224 

9.92  2O2 

9 

4i 

20 

9 

-73997 

9.81  8o3 

0.18  197 

9.92  194 

g 

40 

21 

9 

.74017 

9.81  83i 

o.  18  169 

9.92  186 

9 

39 

22 

9 

.74o36 

9.81  858 

0.18  142 

9.92  177 

38 

23 

9 

.74o55 

19 

9.81  886 

0.18  u4 

9.92  169 

37 

24 

9 

.  74  074 

19 

9.81  913 

27 
28 

0.18  087 

9.92  161 

36 

25 

9 

.74  o93 

9.81  g4 

I 

o.  18  o59 

9.92  i52 

35 

26 

9 

.74n3 

9.81  968 

27 

28 

0.18  o32 

9.92  1  44 

8 

34 

27 

9 

.74  1  32 

19 

9.81  996 

o.  1  8  oo4 

9.92  i36 

9 

33 

28 

9 

.74i5i 

9.82  023 

0.17  977 

9.92  127 

g 

32 

29 

9 

•  74  17° 

19 

9.82  o5i 

0.17  949 

9.92  119 

8 

3i 

30 

9 

•  74  189 

9.82  078 

27 

o.  17  922 

9.92  in 

30 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L. 

Sin. 

d. 

56°  30  . 

PP 

28 

27 

2O 

19 

9 

8 

.1 

2.8 

2.7 

.1 

2.0 

1.9 

i 

0.9 

0.8 

.2 

5-6 

5-4 

.2 

4.0 

3-8 

2 

1.8 

1.6 

•3 

8.4 

8.1 

•3 

6.0 

5-7 

3 

2.7 

2.4 

•4 

II.2 

10.8 

•4 

8.0 

7-6 

4 

3-6 

3.2 

•5 

14.0 

*3»5 

•  5 

10.0 

9-5 

5 

4-5 

4.0 

.6 

16.8 

16.2 

.6 

12.  0 

KM 

6 

5.4 

4-8 

•7 

19.6 

18.9 

•7 

I4.0 

13-3 

7 

6.3 

5-6 

.8 

22.4 

•21.6 

.8 

10.  0 

15-2 

8 

7-2 

6.4 

•9 

25.2            24.3 

•9 

1  8.0                   17.  I 

9 

8.1               7.2 

96 


33°  3O . 


L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L. 

Cos. 

d. 

30 

9 

•  74  189 

9.82078 

28 

0.17  922 

9.92  in 

30 

3i 

9 

.74  208 

9.82  106 

27 

0.17  894 

9.92  1  02 

9 
8 

29 

32 

9 

.74227 

9.82  i33 

28 

0.17  867 

9.92  094 

28 

33 

9 

.74  246 

19 

9.82  161 

27 

0.17  839 

9.92  086 

9 

27 

34 

9 

.74265 

19 

9.82  188 

27 

0.17  812 

9.92077 

s 

26 

35 

9 

.74284 

9.82  2i5 

28 

0.17785 

9.92  069 

25 

36 

9 

.743o3 

X9 

9.82243 

0.17  757 

9.92  060 

9 

24 

19 

27 

8 

37 

9 

.74322 

19 

9.82  270 

28 

0.17  780 

9.92  o52 

8 

23 

38 

9 

.74341 

9.82  298 

o.  17  702 

9.92  o44 

22 

39 

9 

7436o 

19 

9.82  325 

0.17675 

9.92  o35 

9 

8 

21 

40 

9 

.74379 

9.82  352 

28 

0.17  648 

9.92  027 

20 

4i 

9 

.74398 

9.82  38o 

o.  17  620 

9.92  018 

8 

'9 

42 

9 

.744i7 

9.82  407 

_0 

0.175^ 

3 

9.92  oio 

18 

43 

9 

.  74436 

*9 

9.82435 

o.  17  565 

9.92  002 

'7 

jg 

27 

<) 

44 

9 

.74455 

9.82462 

0.17  538 

9.91  993 

8 

16 

45 

9 

.74474 

9.82489 

Oo 

0.17  5u 

9.91  985 

i5 

46 

9.74493 

'9 

9.82  517 

27 

0.17483 

9.91  976 

9 

i4 

47 

9 

74  5i2 

9.82  544 

o.  17  456 

9.91  968 

i3 

48 

9 

7453i 

9.82  671 

o.  17  429 

9.91  959 

(i 

12 

49 

9 

74549 

9.82  599 

o.  17  4oi 

9.91  951 

I  I 

50 

9 

74568 

19 

9.82  626 

0.17  374 

9.91  942 

10 

5i 

9 

74587 

19 

9.82  653 

27 
28 

0.17  347 

9.91  934 

9 

52 

9 

74  606 

*9 

9.82  681 

0.17  319 

9.91  925 

8 

53 

9 

74625 

'9 

9.82  708 

27 

0.17  292 

9.91  917 

7 

54 

9-74644 

'9 

9.82  735 

27 

0.17  265 

9.91  908 

9 
8 

6 

55 

9 

74662 

9.82  762 

0.17  238 

9.91  900 

b 

56 

9 

.74681 

19 

9.82  790 

O.I7  2IO 

9.91  891 

9 

4 

57 

9 

.74  700 

i9 

9.82  817 

27 

0.17  i83 

9.91  883 

0 

3 

58 

9 

•74  7*9 

J9 

9.82  844 

o.  17  i56 

9.91  874 

2 

59 

9 

.74737 

18 

9.82  871 

27 

o.  17  129 

9.91  866 

I 

60 

9 

.74756 

19 

9.82  899 

0.17  101 

9.91  857 

9 

0 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L. 

Sin.     d. 

' 

56°. 

PP 

28 

27 

19 

18 

9 

8 

.1 

2.8 

2.7 

.1 

1.9 

1.8 

.1 

0.9 

0.8 

.2 

5.6 

5-4 

.2 

3.8 

3-6 

.2 

1.8 

1.6 

•3 

8.4 

8.1 

•3 

5-7 

5-4 

•3 

2.7 

2.4 

•4 

II.  2 

10.8 

•4 

7.6 

7.2 

•4 

3-6 

3-2 

,  e 

14.0 

t3-5 

•  5 

9-5 

9.0 

,c 

4-5 

4.0 

.6 

16.8 

16.2 

.6 

11.4 

10.8 

.6 

5-4 

4.8 

•  7 

19.6 

18.9 

•  7 

13-3 

12.6 

•7 

6-3 

5-6 

.8 

22.4 

21.6 

.8 

15-2 

14.4 

.8 

7.2 

6.4 

25.2            24.3 

17.1            16.2 

•9 

8.1 

7.2 

97 


, 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L. 

Cos. 

d. 

0 

9 

•  74756 

9.82  899 

o.  17  101 

9.91  857 

g 

60 

! 

9 

•74775 

19 

9.82  926 

27 

0.17  074 

9.91  849 

59 

2 

9 

.74794 

18 

9.82  953 

0.17  047 

9.91  84o 

g 

58 

3 

9 

.74812 

19 

9.82  980 

28 

0.17  020 

9.91  832 

q 

^>7 

4 

9 

.7483i 

19 

9.83  008 

27 

o.  16  992 

9.91  823 

8 

56 

5 

9 

.74850 

18 

9.83o3< 

o.  16  965 

9.91  815 

55 

6 

9 

,74868 

9.83  062 

0.16  938 

9.91  806 

9 

54 

19 

27 

8 

7 

9 

•  74887 

19 

9.83  089 

28 

0.16  911 

9.91  798 

53 

8 

9 

.74  906 

18 

9-83  117 

o.i6883 

9.91  789 

g 

52 

9 

9 

•  74  924 

9.83  1  44 

27 

o.i6856 

9.91  781 

5i 

10 

9 

.74943 

18 

9.83  171 

0.16  829 

9.91  772 

50 

1  1 

9 

.74  961 

9.83  198 

27 

o.  16  802 

9.91  763 

8 

49 

12 

9 

.74  980 

9.83  225 

0.16  775 

9.91  755 

48 

i3 

9 

.74999 

18 

9.  83  252 

28 

0.16  748 

9.91  746 

9 

8 

47 

i4 

9 

.75  017 

9-83  280 

27 

o.  16  720 

9.91  738 

46 

i5 

9 

.75o36 

18 

9.83  307 

0.16  693 

9.91  729 

45 

16 

9 

.75o54 

9.83334 

27 

o.  16  666 

9.91  720 

9 
8 

44 

ll 

9 

.75  o73 

18 

9.83  36i 

27 

0.16^39 

9.91  712 

43 

18 

9 

.75  091 

9.  83  388 

o.  16  612 

9.91  703 

42 

J9 

9 

.75  1  10 

18 

9.834i5 

0.16  585 

9.91  695 

4i 

20 

9.75  128 

9-83  442 

28 

o.i6558 

9.91  686 

9 

40 

21 

9 

.75  i47 

18 

9.83470 

0.16  53o 

9.91  677 

g 

39 

22 

9 

.75  i65 

9-83497 

0.16  5o3 

9.91  669 

38 

23 

9 

.75  1  84 

r9 

18 

9.83  524 

27 
27 

0.16476 

9.91  660 

y 

9 

37 

24 

9 

.75  202 

9.8355i 

o.  1  6  449 

9.91  65i 

g 

36 

25 

9 

.75  221 

9.83578 

0.16  422 

9.91  643 

35 

26 

9 

.75  23g 

9.836o5 

27 

0.16395 

9.91  634 

y 

34 

27 

.75  258 

'9 
18 

9.  83  632 

27 

o.i6368 

9.91  625 

9 

g 

33 

28 

9 

.75  276 

9.83659 

0.16  34i 

9.91  617 

32 

29 

9 

.75  294 

9.  83  686 

27 

0.16  3i4 

9.91  608 

y 

3i 

30 

9 

.76  3i3 

I9 

9.83  713 

27 

0.16  287 

9.91  599 

y 

30 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L. 

Sin. 

d. 

' 

55°  30  . 

PP 

23 

27 

19 

18 

9 

8 

.1 

2.8 

2.7 

., 

1.0 

1.8 

., 

0.9 

0.8 

.2 

5-6 

5-4 

.2 

3-8 

3-6 

.2 

1.8 

1.6 

•3 

8.4 

8.1 

•3 

5-7 

5-4 

•3 

2.7 

2.4 

•4 

II.  2 

10.8 

•4 

7-6 

7.2 

•4 

3-6 

3-2 

1 

14.0 

16.8 

'I'5 

16.2 

•  5 
.6 

9-5 
11.4 

9.0 

10.8 

:i 

4-5 
5-4 

4.0 
4.8 

•7 

19.6 

i8.q 

7 

13-3 

12.6 

.7 

6-3 

5-6 

.8 

22.4 

21.6 

.8 

15-2 

14.4 

.  8 

7.2 

6.4 

•  0 

25.2            24.3 

17.1            16.2 

8.1              7.2 

98 


34°  30' 


! 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

30 

9 

.75  3i3 

18 

9.83 

7i3 

27 
28 
27 
27 
27 
27 
27 
27 
27 
27 

27 
27 
27 
27 
27 
27 
27 
27 
27 

27 

26 
27 
27 
27 

27 
27 
27 
27 
27 
27 

0. 

i6287 

9.91 

599 

30 

3i 

32 

33 

34 
35 
36 

37 
38 
39 

9 
9 

9 

9 
9 

9 

9 
9 
9 

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.75  368 

.75386 

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.75459 

18 

18 

'9 
18 

18 
18 

19 

18 

18 

if 

18 
if 
if 

18 
18 
if 

18 
if 

if 

18 
if 
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18 
if 

18 

9.83  740 
9.83  768 
9.  83  795 

9-83  822 
9.83  849 
9.83  876 

9.83  903 
9.83  930 
9.83957 

0. 
0. 

o. 

o. 
o. 

0. 

o. 
o. 
o. 

16  260 

16  232 

16  2o5 

16  178 
16  i5i 
1  6  124 

16  097 
1  6  070 
i6o43 

9.91 
9.91 
9.91 

9.91 
9.91 
9.91 

9.91 
9.91 
9.91 

59i 

582 
573 

565 
556 
547 
538 
53o 

521 

9 
9 
8 

9 
9 
9 
8 

9 

29 

28 
27 

26 

25 

24 

23 
22 
21 

40 

9.75496 

9.83 

984 

o. 

16  016 

9.91 

5l2 

8 
9 
9 
9 
8 

9 

9 
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9 
8 

9 
9 
9 
9 
8 

9 
9 
9 
9 

9 

20 

4i 

42 

43 

44 
45 
46 

47 

48 

49 

9.755i4 
9.75533 
9.7555i 

9.  75  569 
9.75587 
9.75  6o5 

9.75  624 
9.75  642 
9.75  660 

9.84  01  1 
9.84o38 
9.84  065 

9.84  092 
9.84  119 
9.84  i46 

9.84  i73 
9.84  200 

9.84  227 

o. 

0. 
0. 

o. 

0. 
0. 

0. 
0. 

o. 

i5  989 
1  5  962 
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1  5  908 
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1  5  854 

i5  827 
1  5  800 
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9.91 
9.9i 
9.91 

9.91 
9.91 
9.91 

9.91 
9.91 
9.91 

5o4 

495 

486 

477 
469 
46o 

45i 
442 
433 

19 
18 

17 

16 
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12 
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50 

9 

.75  678 

9-84 

254 

o. 

i5746 

9.91 

425 

10 

5i 

52 

53 

54 
55 
56 

57 
58 
59 

9.70  696 
9.75  714 
9.75  733 

9.7575i 
9.75  769 
9.75787 

9.7^805 
9.75  823 
9.75  84  1 

9.84  280 
9.84  3o7 
9.84334 

9.8436i 
9.84388 
9-844i  5 

9.84442 
9.84469 
9.84496 

0. 
0. 
0. 

0. 
0. 
0. 

o. 
o. 

0. 

1  5  720 
i5693 
1  5  666 

i5639 
i5  612 

i5558 
i553i 
i55o4 

9.91 
9.91 
9.91 

9.91 
9.91 
9.91 

9.91 
9.91 
9.91 

4i6 
407 
398 

389 
38i 

372 

363 
354 
345 

9 

8 

7 

6 
5 

4 
3 

2 

I 

60 

9.75859 

9-84 

523 

0. 

i5  477 

9.91 

336 

0 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L. 

Tang. 

L.  Sin. 

d. 

55°. 

PP 

.2 

•3 

•4 

•9 

28 

27 

26 

19 

if 

.2 

•3 

•4 

9           8 

2.8 

5-6 
8.4 

II.  2 
14.0 

1  6.  8 
19.6 

22-4 

2.7 

10.8 

II;  2 

18.9 

21.6 

24-3 

2.6 

10.4 
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15-6 

18.2 

20.8 

23-4 

.1           1.9 

.2           3-8 
•3            5-7 

•4            7-6 
•5            9-5 
.6          11.4 

•7          J3-3 
.8          15-2 

.q           17.1 

1.8 
3-6 

5-4 

7.2 
9.0 
10.8 

12.6 

14.4 

16.2 

0.9            0.8 
1.8            1.6 
2-7             2.4 

3-6            3-2 
4-5            4-o 
5-4            4-8 

6-3            5.6 
7.2            6.4 

8.1             7.2 

99 


35°. 


/ 

L.  Sin.       d. 

L.  Tang. 

d. 

L.  Cotg. 

L. 

Cos. 

d* 

0 

9 

.75859 

18 

9.84  5-23 

o.i5  477 

9.91  336 

60 

I 

9 

•75877 

18 

9.84  550 

26 

o.  i5  45o 

9.91  328 

59 

2 

9 

.75895 

18 

9.84576 

o.  i5  424 

9.91  3i9 

58 

3 

9.75913 

9.846o3 

o.i  5  397 

9.91  3io 

9 

57 

27 

9 

4 

9 

.7593i 

18 

9.8463o 

27 

o.  i5  37o 

9.91  3oi 

56 

5 

9 

.75949 

18 

9.84657 

o.i5343 

9.91  292 

55 

6 

9 

•75967 

18 

9.84684 

27 

o.i5  3i6 

9.91  283 

y 

9 

54 

7 

9 

.75985 

18 

9.84  7n 

27 

o.i5  289 

9.91  274 

8 

53 

8 

9 

.76  oo3 

9.84738 

o.i5  262 

9.91  266 

52 

9 

9 

.76  O2I 

9-84  764 

o.i5  236 

9.91  257 

9 

5i 

10 

9 

.76039 

18 

9.84  79i 

27 

o.i  5  209 

9.91  248 

50 

ii 

9 

.76057 

18 

9.84818 

27 

o.i5  182 

9.91  239 

g 

49 

12 

9 

.76075 

18 

9.84845 

o.i5  i55 

9.91  23o 

48 

i3 

9 

.76  o93 

18 

9.84872 

27 

o.i5  128 

9.91  221 

Q 

47 

i4 

9 

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18 

9.84899 

26 

o.  i5  101 

9.91  212 

9 

46 

i5 

9 

.76  I29 

9.84  925 

o.i5  or 

75 

9.91  2o3 

45 

16 

9 

.76l46 

18 

9.84  952 

27 

o  .  1  5  o48 

9-91   194 

9 

44 

'7 

9 

.76164 

18 

9.84979 

27 

o.  1  5  02  1 

9.91  i85 

9 

43 

18 

9 

.76  l82 

18 

9.85  006 

0.14994 

9.91  176 

42 

19 

9 

.76  200 

9.85o33 

26 

o.  1  4  967 

9.91  167 

4i 

20 

9 

.76  218 

9.85  059 

o  .  1  4  94  1 

9.91  i58 

40 

21 

9 

.76  236 

17 

9.85  086 

27 

o.  i4  914 

9.91  149 

8 

39 

22 

9 

.76253 

9.85  n3 

0.14887 

9.91  i4i 

38 

23 

9 

.76271 

18 

9.85  i4o 

26 

o.  i4  860 

9.91  i32 

9 

37 

24 

9 

.76  289 

18 

9.85  166 

27 

o.i4834 

9.91    123 

9 

36 

25 

9 

.763o7 

9.85  193 

o.  i4  807 

9.91  n4 

35 

26 

9 

.  76  324 

18 

9.85  220 

27 

o.  14  780 

9.91  105 

9 

34 

27 

9 

.76342 

18 

9.85  247 

26 

o.i4  7^ 

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9.91  096 

9 

33 

28 

9 

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9.85  273 

o.i4  727 

9.91  087 

32 

29 

9 

.76  378 

9.85  3oo 

o.  i4  700 

9.91  078 

9 

3i 

30 

9 

.76395 

I7 

9.  85  327 

o.  i4  673 

9.91  069 

30 

L.  Cos.      d. 

L.  Cotg. 

d. 

L.  Tang. 

L. 

Sin. 

d. 

' 

54°  3O  . 

PP 

27 

26 

18 

17 

9                8 

.1 

2-7 

2.6 

.1 

1.8 

r-7 

.1 

0.9             0.8 

.2 

5-4 

5-2 

.2 

3-6 

3-4 

.2 

1.8              1.6 

•3 

8.1 

7.8 

3 

5-4 

5-1 

•3 

2.7              2.4 

•4 

10.8 

10.4 

•4 

7.2 

6.8 

•4 

3-6              3-2 

[11 

13.0 

15-6 

9.0 
10.8 

8-5 

10.2 

•5 
.6 

4-5              4-° 
5-4              4-8 

•  7 

18.9 

18.2 

-7 

12.6 

ii.  9 

•7 

6.3              5-6 

.8 

21.6 

20.8 

.8 

14.4 

13-6 

.8 

7.2              6.4 

-9 

24--?                    2}.  4 

•9 

16.2             15.3 

• 

8.1               7.2 

35°  3O 


r 

L.  Sin.       d. 

L.  Tang. 

d. 

L.  Cotg. 

L. 

Cos. 

d. 

30 

9 

.76  395 

9.85  327 

27 
26 
27 
27 
26 
27 
27 
26 
27 
27 

26 
27 
27 
26 
27 
27 
26 
27 
27 
26 

27 
26 
27 
27 
26 
27 
26 
27 
27 
26 

o.  14  673 

9.91  069 

9 
9 
9 
9 
ro 

9 
9 
9 
9 

30 

3i 

32 

33 

34 
35 
36 

3? 
38 
39 

9 
9 
9 

9 
9 
9 

9 
9 
9 

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.7643i 
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.76  519 
.76537 
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18 
17 

18 
18 

'7 
18 
18 
17 

9.  85  354 
9.  85  38o 
9.85407 

9.85434 
9.85  46o 
9.  85  487 

9.85  5i4 
9.85  54o 
9.85567 

o.  i4  646 
o.  i4  620 
o.  i4  5g3 

o.i4566 
o.  i4  54o 
o.i4  5i3 

o.i4486 
0.14  46o 
o.i4433 

9.91  060 
9.91  o5i 

9.91  042 

9.91  o33 
9.91  oa3 
9.91  oi4 

9.91  oo5 
9.90996 
9.90987 

29 

28 
27 

26 

25 

24 

23 
22 
21 

40 

9 

.76572 

9.85  594 

o.i4  4o6 

9.90978 

9 
9 
9 
9 
9 
9 
9 
9 

10 

9 

9 
9 
9 
9 
9 

10 

9 
9 
9 
9 

20 

4i 

42 

43 

44 
45 
46 

47 
48 

49 

9.76  590 
9.76  607 
9.76  625 

9.76  642 
9.76  660 
9.76677 

9.76695 
9.76  712 
9.76  73o 

17 

18 

'7 
18 

17 

18 

»7 

18 

9.85  620 
9.85  647 
9.85  674 

9-85  700 
9.85  727 
9.85754 

9-85  780 
9.85  807 
9.85  834 

o.i4  38o 
o.i4353 
o.i4  326 

o.  1  4  3oo 
o.i4  273 
o.i4  246 

0.  l4  220 

o.i4  193 
0.14  166 

9.90969 
9.90  960 
9.9o95i 

9.90  942 
9.90  933 
9.90  924 

9.90915 
9.90  906 
9.90  896 

J9 

18 

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16 
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12 
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50 

9 

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9.90  887 

10 

5i 

52 

53 

54 
55 
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59 

9 
9 
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9 
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9 
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76765 
76  782 
76  800 

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76  887 
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17 

18 

17 

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9.85  887 
9.85  913 
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9.85  967 
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9.86  020 

9.86  o46 
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o.i4  n3 
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o.i3  954 
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9.90  878 
9.90  869 
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9.90  85i 
9.90  842 
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9.90  805 

9 

8 

7 

6 

5 

4 

3 

2 
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9 

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9.86  126 

o.i  3  874 

9.90796 

L.  Cos. 

d. 

L.  Cotg.      d. 

L.  Tang. 

L.  Sin. 

d. 

t 

54°. 

PP 

.2 

•3 
•4 

:5 

27 

26 

18 

17 

.2 

•3 
•4 

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10 

9 

2.7 

5-4 
8.1 

10.8 

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16.2 

18.9 

21.6 

a*  T, 

2.6 
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7-8 

10.4 
13.0 
15-6 

18.2 

20.8 

23.4 

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1.8 
3-6 
5-4 

7.2 
9.0 
10.8 

12.6 

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3-4 
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6.8 
8-5 

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II-9 
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I.O 

2.0 

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4.0 

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6.0 

7.0 
8.0 

9.0 

ti 

2.7 

3-6 
4-5 
5-4 

6-3 
tj 

101 


36C 


1 

L.Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L. 

Cos.    i  d. 

0 

9 

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9.86  126 

27 
26 
27 
26 
27 
26 
27 
26 
27 
27 

26 
27 
26 
27 
26 
27 
26 
26 
27 
26 
27 
26 
27 
26 
27 
26 
27 
26 
26 
27 

o.i3  874 

9.90  796 

9 

10 

9 
9 
9 
9 

10 

9 
9 
9 

10 

9 
9 
9 
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9 
9 
9 

10 

9 

60 

I 

2 

3 

4 
5 
6 

8 
9 

9 
9 
9 

9 
9 
9 

9 
9 
9 

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18 
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17 
18 

17 
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9.86  i53 
9.86  179 
9.86  206 

9.86  232 
9.86  259 
9.86285 

9.86  3i2 
9.86338 
9.86  365 

o.  i3  847 
o.i3  821 
o.i  3  794 

o.i3  768 
o.i3  74i 
o.i3  715 

o.i3688 
o.i3  662 
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9.90787 
9.90777 

9  .90  768 

9.90759 
9.90750 
9.90  741 

9.90731 
9.90  722 
9.90713 

59 

58 
57 
56 
55 
54 

53 

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10 

9 

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18 

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9.86  392 

o.i  3  608 

9.90  704 

50 

ii 

12 

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16 

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9 
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9 
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9.86418 

9-86445 
9.86471 

9.86498 
9.  86  524 
9.8655i 

9.  86577 
9.86  6o3 
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9.90  694 
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9.90  667 
9.90  657 
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9.90  620 

49 

48 

47 
46 
45 
44 

43 

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20 

9 

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9.86656 

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9.90  61  1 

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21 
22 
23 

24 
25 

26 

27 
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9 
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9.86  709 
9.86  736 

9.86  762 
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9.86842 
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o.i3  185 

o.i3i58 
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9.90  602 
9.90  592 
9.90  583 

9.90  574 
9.  90  565 
9.90  555 

9.90  546 
9.90  537 
9.90  527 

10 
9 
9 
9 
10 

9 
9 
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39 
38 
37 
36 
35 
34 

33 

32 

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30 

9 

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9.86  921 

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9.90  5i8 

y 

30 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L. 

Sin. 

d. 

53°  3O  . 

PP 

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•3 
4 

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9 

27 

26 

18 

17 

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2 

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10 

9 

2.7 

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10.8 

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18.9 

21.6 

24.3 

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20.8 

23.4 

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6.8 
8-5 

10.2 

II.Q 

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15.3 

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2.0 

3-° 

4.0 

5-° 
6.0 

7.0 
8.0 
9.0 

0.9 
1.8 
2.7 

3-6 
4-5 
5-4 

6-3 

g 

102 


30°  30 . 


! 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

30 

9 

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9.86  921 

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9.90  5i8 

30 

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17  . 

9.86  947 

27 

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32 

9 

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9.86  974 

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9.9 

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28 

33 

9 

77490 

17 

9.87  ooo 

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9.90490 

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27 

17 

27 

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34 

9 

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9.87027 

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35 

9 

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0.12  947 

9.90471 

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36 

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9.90434 

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21 

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9.90424 

20 

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9.87  211 

9.87  238 

27 

0.12  789 
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9.87  264 

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L.  Cos. 

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d.     L.  Tang. 

L. 

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53°. 

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37 


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L.  Sin. 

d. 

L.  Tang. 

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L.  Cotg. 

L.  Cos. 

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0 

9 

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39 

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52°  30  . 

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L.  Sin. 

d. 

L.  Tang. 

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30 

9.73445 

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d. 

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i.i 

1.0 

.2 

5-2 

5-o 

.2 

3-2 

3-° 

.2 

2.2 

2.O 

•3 

7.8 

7-5 

•3 

4.8 

4-5 

•3 

3-3 

3-° 

•4 

10.4 

IO.O 

•4 

6.4 

6.0 

•4 

4.4 

4.0 

j 

13.0 
15-6 

12.5 
15-0 

1 

8.0 
9.6 

7-5 
9.0 

:J 

Ii 

C 

•  7 

18.2 

17-5 

•7 

II.  2 

10.5 

•7 

7-7 

7.0 

.8 

20.8 

20.  o 

.8 

12.8 

12.  0 

.8 

8.8 

8.0 

23.4          22.5              .9 

14.4          13.5 

•9 

9.9              9.0 

107 


39°. 


! 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L. 

Cos.      d. 

0 

9 

.79887 

16 

9.90  837 

26 

0.09  i63 

9  .  89  o5o 

60 

i 

9 

.79903 

15 

9.90  863 

26 

0.09  1  37 

9.89  o4o 

IO 

59 

2 

9 

•799'8 

16 

9.90  889 

0.09  in 

9.89  o3o 

58 

3 

9 

.79934 

16 

9.90  914 

26 

0.09  086 

9.89  020 

ii 

D7 

4 

9 

.79950 

15 

9.90  940 

26 

0.09  060 

9.89  009 

JO 

56 

5 

9 

.79965 

16 

9.90  96^ 

) 

26 

0.09  o34 

0.88999 

55 

6 

9 

.79981 

15 

9.90992 

26 

o  .  09  008 

9.88989 

II 

54 

7 

9 

.79996 

16 

9.91  018 

25 

0.08  982 

9.8 

8978 

10 

53 

8 

9 

.80  012 

9.91  o4- 

26 

0.08  957 

9.88  968 

52 

9 

9 

.80  O27 

16 

9.91  069 

26 

0.08  9: 

h 

9.88  958 

5i 

10 

9 

.80043 

15 

9.91  ogS 

26 

0.08  905 

9.88948 

50 

ii 

9 

.8oo58 

16 

9.91    121 

26 

0.08  879 

9.88  937 

49 

12 

9 

.80  074 

9.91  147 

0.08  853 

9.88  927 

48 

i3 

9 

.80089 

16 

9.91  172 

26 

0.08  828 

9.88917 

II 

47 

i4 

9 

.80  105 

15 

9.91  198 

26 

0.08  802 

9.88  906 

46 

i5 

9 

.80  I2O 

16 

9.91   224 

26 

0.08  776 

9.88  896 

45 

16 

9 

.80  i36 

15 

9.91   250 

26 

0.08  750 

9.8 

8  886 

II 

44 

17 

9 

.80  i5i 

15 

9.91  276 

25 

0.08  724 

9.8 

8  875 

43 

18 

9 

.80  166 

16 

9.91  3oi 

26 

0.08  699 

9.88  865 

42 

19 

9 

.80  182 

9.91  32^ 

r 

26 

0.08  673 

9.88  85s 

4i 

20 

9 

.80  197 

16 

9.91  353 

26 

0.08  647 

9.88844 

40 

21 

9 

.80  2i3 

15 

9.91  379 

25 

0.08  621 

9.8 

8834 

39 

22 

9.80228 

16 

9.91  4o^ 

i 

26 

0.08  596 

9.88  824 

38 

23 

9 

.80  244 

15 

9.91  43o 

26 

0.08  570 

9.88  8i3 

IO 

37 

24 

9 

.80259 

15 

9.91  456 

26 

0.08  544 

9.88  8o3 

36 

25 

9 

.80274 

9.91  482 

0.08  5i8 

9.88  793 

35 

26 

9 

.80  290 

9.91  507 

25 

0.08  493 

9.88  782 

34 

15 

26 

IO 

27 

9 

.80  3o5 

15 

9.91  533 

26 

0.08  467 

9.88  772 

33 

28 

9 

.80  320 

9.91  559 

0.08  44  1 

9.88  761 

32 

29 

9 

.80  336 

9.91  58s 

0.08  4i5 

9.8 

875i 

3  1 

30 

9 

.8o35i 

9.91  610 

25 

0.08  390 

9.8 

74i 

30 

L.  Cos. 

d. 

L.  Cotg.  |  d. 

L.  Tang. 

L. 

Sin. 

d. 

50°  30  . 

PP 

26 

25 

16 

15 

ii 

10 

.1 

2.6 

2.5 

.1 

1.6 

'•5 

.1 

i.i 

I.O 

>2 

5-2 

5.0 

.2 

3-2 

3-° 

.2 

2.2 

2.O 

•3 

7.8 

7-5 

•3 

4.8 

4-5 

•3 

3-3 

3-° 

•4 

10.4 

IO.O 

-4 

6.4 

6.0 

•4 

4-4 

4.0 

.5 

13.0 

12.5 

•  5 

8.0 

7'  5 

5-5 

5«o 

.6 

15.6 

15.0 

.6 

9.6 

9.0 

.6 

6.6 

6.0 

•I 

18.2 

20.8 

17-5 

20.  0 

:l 

II.  2 
12.8 

10.5 

12.0 

:! 

11 

7.0 
8.0 

•Q 

23.4           22.5 

•9 

14.4 

13.5 

1 08 


39°  3O . 


« 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

30 

9 

.8o35i 

9.91  610 

26 
26 
26 
25 
26 
26 
26 
25 
26 
26 

25 
26 
26 
26 
25 
26 
26 
25 
26 
26 

25 
26 
26 

25 
26 
26 
25 
26 
26 
25 

0.08  390 

9.88  741 

30 

3i 

32 

33 

34 
35 
36 

3? 
38 
39 

9 
9 
9 

9 
9 
9 

9 
9 
9 

.80  366 
.80  382 
.80  397 

.80412 
.80428 
.8o443 

.80  458 
.8o473 
.80489 

16 
i5 
15 
16 

15 
i5 
15 
16 

15 

9.91  636 
9.91  662 
9.91  688 

9.91  713 
9.91  739 
9.91  765 

9.91  791 
9.91  816 
9.91  842 

0.08  364 
0.08  338 
0.08  3i2 

0.08  287 
0.08  261 
0.08  235 

0.08  209 
0.08  i84 
0.08  i58 

9.88  730 
9.88  720 
9.88  709 

9.88  699 
9.88688 
9.88678 

9.88668 
9.88657 
9.88  647 

10 

II 

IO 

II 

10 
IO 

II 

10 

29 

28 
27 

26 

25 
24 

23 
22 
21 

40 

9 

80  5o4 

9.91  868 

0.08  i32 

9.88636 

20 

4i 

42 

43 

44 
45 
46 

4? 
48 

49 

9 
9 
9 

9 
9 
9 

9 
9 
9 

.80  519 
.80  534 
.80  550 

.80565 
.80  58o 
.80595 

.80  610 
.80625 
.8o64i 

15 
i5 

16 

15 

15 
15 
15 
IS 
16 
15 

i5 

15 

15 

»5 
»5 
15 

16 

15 
15 

IS 

9.91  893 
9.91919 
9.91  945 

9.91971 
9.91  996 
9.92  022 

9.92  o48 
9.92  073 
9.92099 

0.08  107 
0.08  081 
o.o8o55 

0.08  029 
o  .  08  oo4 
0.07  978 

0.07  952 
0.07  927 
0.07  901 

9.88  626 
9.886i5 
9.88  605 

9.88  594 
9.88  584 
9.88673 

9.88563 
9.88  552 
9.88  542 

II 

10 

II 

IO 

II 

IO 

11 

10 

II 

IO 

II 
II 

IO 

II 

IO 

II 

10 

II 
II 

i9 

18 

!? 

16 
i5 
i4 
i3 

12 
II 

50 

9 

.80  656 

9.92  125 

0.07  875 

9.88  53i 

10 

9 

8 

7 

6 
5 

4 
3 

2 

I 

5i 

52 

53 

54 
55 
56 

57 
58 
59 

9.80671 
9.80686 
9.80  701 

9.80  716 
9.80  731 
9.80  746 

9.80  762 
9.80777 
9.80  792 

9.92  i5o 
9.92  176 
9.92  202 

9.92  227 
9.92  253 
9.92279 

9.92  3o4 
9.92  33o 
9.92  356 

0.07  850 
0.07  824 

0.07  798 

0.07  773 
0.07747 

0.07  721 

0.07  696 
0.07  670 
0.07  644 

9.88  521 

9.88  5io 
9.88499 

9.88489 
9.88478 
9.88468 

9.88457 
9-88447 
9.88436 

60 

0 

.80807 

9.92  38i 

0.07  619 

9.88425 

0 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L. 

Sin. 

d. 

' 

5O°. 

PP 

.2 

•3 

•4 
•5 
.6 

•  7 
.8 

26 

25 

16 

15 

.1 

.2 

•3 
•4 

9 

II                    10 

2.6 

5-2 

7.8 

10.4 

13.0 
156 

18.2 

20.8 

2V4 

2.5            .1 

5.0                          .2 

7-5                    -3 

10.0                    .4 
12-5                  .5 
15.0                  .6 

17-5                    -7 

20.0                          .8 

22.5                   .9 

1.6 

43:s 
6.4 

8.0 
9.6 

II.  2 

12.8 

M-4 

1  5 
3-° 
45 

6.0 
7-5 
9.0 

10.5 

12.  0 

13-5 

I.I                      1.0 
2.2                    2.0 

3-3              3-o 

4-4              4-° 
5-5              5-° 
6.6              6.0 

7-7              7-o 
8.8              8.0 

9.9              9.0 

109 


40< 


/ 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L. 

Cos.     d. 

0 

9 

.80807 

15 
15 
15 
15 
15 
15 
15 
15 
15 

9.92  38i 

26 
26 

25 
26 
26 
25 
26 
26 
25 
26 

25 
26 
26 

25 
26 
26 
25 
26 
25 
26 

26 
25 
26 
25 
26 
26 

25 
26 

25 
26 

0.07  619 

9.88425 

10 
ii 

10 

II 
II 

10 

II 
II 

10 

II 
II 

10 

II 
II 

10 

II 
II 

10 

II 

60 

I 

2 

3 

4 
5 
6 

8 
9 

9 

9 
9 

9 
9 
9 

9 
9 
9 

.80  822 
.8o837 
.80  852 

.80  867 
.80882 
.80897 

.80  912 
.80  927 
.80  942 

9.92  407 
9.92433 
9.92458 

9.92484 
9.92  5io 
9.92  535 

9.92  56i 
9.92  587 
9.92  612 

0.07  5g3 
0.07  567 
0.07  542 

0.07  5i6 
0.07  490 
0.07465 

0.07  439 
0.07  4i3 
0.07  388 

9.88415 
9.88404 
9.88  394 

9.88  383 
9.88  372 
9.88  362 

9.8835i 
9.88  34o 
9.8833o 

59 
58 
57 
56 
55 
54 
53 

52 

5i 

10 

9 

.80957 

J5 

9.92  638 

0.07  362 

9.8 

8  3i9 

50 

1  1 

12 

i3 

i4 
i5 
16 

i? 

18 

J9 

9 
9 
9 

9 

9 
9 

9 
9 
9 

.80  972 
.80  987 

.8l  002 

.8l  017 
.8l  032 

.81  047 

.81  061 
.81  076 
.81  091 

15 
i5 
15 
'5 
15 
*4 
15 
15 
15 

15 
i5 
15 
i5 
14 
i5 
15 
i5 
15 
'4 

9.92  663 
9.92  689 
9.92715 

9.92  740 
9.92  766 
9.92792 

9.92817 
9.92  843 
9.92  868 

0.07  337 
0.07  3i  i 
0.07  285 

0.07  260 
0.07  234 
0.07  208 

0.07  i83 
0.07  157 
0.07  i32 

9.88  3o8 
9.88298 
9.88  287 

9.88  276 
9.88266 
9.88  255 

9.88  244 
9.88  234 

9.88  223 

49 

48 

47 

46 
45 
44 

43 

42 

4i 

20 

9 

.81  1  06 

9.92  89^ 

1 

0.07  106 

9.88  212 

40 

21 
22 
23 

24 
25 
26 

27 
28 
29 

9 
9 
9 

9 
9 
9 

9 

9 
9 

.8l    121 

.81  i36 

.81  i5i 

.81  166 
.81  1  80 
.81  i95 

.8l  210 

.81  225 
.81  240 

9.92  920 
9.92  945 
9.92971 

9.92996 

9.93  022 
9.93048 

9.93o73 
9.93099 
9.93  124 

0.07  080 
0.07  o55 
0.07  029 

0.07  oo4 
0.06  978 
0.06  952 

0.06  927 
0.06  901 
0.06  876 

9.88  201 
9.88   191 
9.88  l8o 

9.88  169 

9.88  i58 
9.88  i48 

9.88  137 
9.88  126 
9.88  u5 

IO 

II 
II 
II 

10 

II 
II 
II 

39 
38 
37 
36 
35 
34 
33 

32 

3i 

30 

9 

.81  254 

9.93  150 

0.06  85o 

9.8 

8  105 

30 

L.  Cos.      d. 

L.  Cotg. 

d. 

L.  Tang. 

L. 

Sin.    }d. 

r 

49°  30  . 

PP 

.1 

.2 

•3 
•4 

:l 

•  7 
.8 

•9 

26 

25 

15 

M 

.1 
.2 

•3 

•4 
•5 
.6 

•  7 

.8 

ii 

IO 

2.6 
5-2 

7.8 

10.4 
13.0 

15.6 

18.2 

20.8 

23-4 

2.5 
5-0 
7-5 

10.0 

12.5 
15-0 

'7-5 

20.0 

22.5 

.1 

.2 

•  3 
•4 

•  7 
.8 

r.S 

3-o 
4-5 

6.0 
7-5 
9.0 

10.5 

12.0 

*3-5 

M 

2.8 
4-2 

5.6 

7.0 

8.4 
9-8 

II.  2 

i.i 

2.2 

3-3 

4-4 
5-5 
6.6 

7-7 
8.8 

9-9 

1.0 
2.O 

3-0 

4.0 

5-° 
6.0 

7.0 
8.0 
9.0 

40°  30 ' 


/ 

L.  Sin.       d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

30 

9.81  254 

9.93  150 

25 
26 
26 
25 
26 
25 
26 
25 
26 
26 

25 
26 
25 
26 

25 
26 

25 

26 
26 

.  0.06  85o 

9.88  105 

ii 
ii 
ii 
ii 

IO 

II 
II 
II 
II 
II 

II 

10 

II 
II 
II 
II 
II 
II 
II 
II 

30 

3i 

32 

33 

34 
35 
36 

37 
38 
39 

9.81  269 
9.81  284 
9.81  299 

9.81  3i4 
9.81  328 
9.81  343 

9.81  358 
9.81  372 
9.81  387 

15 

15 

15 
15 
M 

15 
15 
*4 

15 

9.93  i75 

9.93  2OI 
9.93  227 

9.93  252 
9.93278 

9.93  3o3 

9.93  329 
9.93354 
9.93  38o 

0.06  825 
0.06  799 
0.06  773 

0.06  748 
0.06  722 
0.06  697 

0.06  671 
0.06  646 
0.06  620 

9.88  094 
9.88o83 
9.88  072 

9.88  061 
9.88  o5i 
9.88  o4o 

9.88  029 
9.88  018 
9.88  007 

29 

28 
27 

26 

25 
24 

23 
22 
21 

40 

9 

.81  402 

9.93  4o6 

0.06  5g4 

9.87  996 

20 

4i 

42 
43 

44 
45 
46 

4? 
48 

49 

9 
9 
9 

9 
9 
9 

9 
9 
9 

.81  417 

.81  43i 
.81  446 

.81  46  1 
.8i4?5 
.81  490 

.81  505 

.81  519 
81  534 

M 
15 
15 
M 
15 
15 
'4 
15 

9.9343i 
9.93457 
9.93  482 

9.93  5o8 
9.93  533 
9.93559 

9.93  584 
9.93  610 
9.93636 

0.06  569 
o.o6543 
0.06  5i8 

0.06  492 
0.06  467 
0.06  44  1 

0.06  4i6 
0.06  390 
0.06  364 

9.87  985 

9-87975 
9.87  964 

9.87953 
9.87  942 
9.8793i 

9.87  920 
9.87909 
9.87  898 

J9 

18 

!7 

16 
i5 

i4 

i3 

12 
II 

50 

9.81  549 

9.93  661 

26 

25 
26 
25 
26 
25 
26 
25 
26 
25 

0.06  339 

9.87  887 

10 

5i 

52 

53 

54 
55 
56 

5? 
58 
59 

9 

9 
9 

9 
9 
9 

9 

9 
9 

81  563 

81  578 
81  592 

81  607 
81  622 
.81  636 

.81  65i 
81  665 
.81  680 

15 
14 
IS 
IS 
14 
»5 
14 
15 

9.93  687 
9.93  712 
9.93738 

9.93763 
9.93789 
9.93  8i4 

9.93  84o 
9.93865 
9.93  891 

0.06  3  1  3 
0.06  288 
0.06  262 

0.06  237 

0  .  06  2  I  I 

0.06  186 

0.06  1  60 
0.06  135 
0.06  109 

9.87877 
9.87  866 
9.87855 

9.87844 
9.87833 
9.87  822 

9.87  811 
9.87  800 
9.87789 

II 

II 
II 
II 
II 
II 
II 
II 
II 

9 

8 

7 

6 
5 
4 
3 

2 

I 

60 

.81  694 

9.93  916 

0.06  o84 

9.87778 

0 

L.  Cos.       d. 

L.  Cotg 

d.     L.  Tang. 

L. 

Sin. 

d. 

f 

49°. 

PP 

.1 

.2 

•3 
4 

:J 

26 

«5 

15 

14 

.2 

•3 

•4 
•5 
.6 

:5 

ii 

10 

2.6 

5.  2 

7.8 

10.4 
13.0 
'5.6 

18.2 

20.8 

23-4 

2-5 

5-o 
7-5 

IO.O 

12.5 
15-0 

17-5 

20.  o 

.1 

.2 

•3 

•4 
•  5 
.6 

:i 

i-5 
3-o 
4-5 

6.0 
7-5 
9.0 

10.5 

12.0 

n-5 

1.4 

2.8 

4.2 

5-6 
7.0 
8.4 

9.8 

II.  2 

1.  1 

2.2 

3-3 

4-4 
5-5 
6.6 

7-7 
8.8 

9.9 

J.O 
2.0 

3-o 

4.0 

5-o 
6.0 

7.0 
8.0 

41°. 


> 

L.  Sin.       d. 

L.  Tang.     d. 

L.  Cotg. 

L.  Cos. 

d. 

0 

9.81  694 

9.93  916 

26 

0.06  084 

9.87778 

60 

I 

9.81  709 

14 

9.93  942 

25 

0.06  o58 

9.87767 

5Q 

2 

9.81  723 

9.93967 

26 

0.06  o33 

9.87756 

58 

3 

9.81  738 

9.93993 

25 

0.06  007 

9.87745 

57 

4 

9.81  752 

15 

9  .  94  o  1  8 

26 

o.o5  982 

9.87734 

56 

5 

9.81  767 

9.94  o44 

0.  05956 

9.87723 

55 

6 

9.81  781 

15 

9.94069 

26 

o.o5  931 

9.87712 

54 

7 

9.81796 

14 

9.94095 

25 

o.o5  905 

9.87  701 

53 

8 

9.81  810 

9.94  I2O 

26 

o.o5  880 

9.87  690 

52 

9 

9.81  825 

9.94  i46 

o.o5854 

9.87679 

5i 

10 

9.81  839 

15 

9-94  171 

25 
26 

o.o5  829 

9.87668 

50 

ii 

9.81  854 

9.94197 

25 

o.o5  8o3 

9.87657 

49 

12 

9.81  868 

9.94  222 

26 

o.o5  778 

9.87  646 

48 

i3 

9.81  882 

15 

9.94  248 

25 

o.o5  752 

9.87635 

M 

47 

i4 

9 

.81  897 

14 

9.94  273 

26 

o.o5  727 

9.87624 

46 

i5 

9.81  911 

9-94299 

o.o5  701 

9.87613 

45 

16 

9 

.81  926 

9.94324 

26 

o.o5  676 

9.87  601 

44 

17 

9 

.81  940 

15 

9.94350 

25 

o.o5  65o 

9.87590 

TI 

43 

18 

9 

.81955 

9.94375 

26 

o.o5  625 

9-87579 

42 

'9 

9 

.81  969 

9  .  94  4o  i 

o.o5  599 

9.87  568 

4i 

20 

9 

.81  983 

9.94426 

26 

o.o5  574 

9.87557 

40 

21 

9.81998 

14 

9.94  452 

25 

o.o5  548 

9.87  546 

39 

22 

9 

.82  OI2 

9.94477 

26 

o.o5  523 

9.87535 

38 

23 

9 

.82  026 

9.94  5o3 

o.o5  497 

9.87  524 

37 

15 

25 

24 

9 

.82  o4i 

9.94  528 

26 

o.o5  472 

9.87613 

12 

36 

25 

9 

.82  065 

9.94554 

o.o5  446 

9.87  5oi 

35 

26 

9 

.82  069 

I4 

9.94579 

o.o5  421 

9.87490 

34 

15 

25 

27 

9 

.82  o84 

9  .  94  6o4 

26 

o.o5  396 

9.87479 

II 

33 

28 

9 

.82  098 

9.94  63o 

o.o5  370 

9.87468 

32 

29 

9 

.82   112 

14 

9.94655 

26 

o.o5  34s 

9-87457 

3i 

30 

9 

.82  126 

9.94  681 

o.o5  319 

9.87446 

30 

L.  Cos.      d. 

L.  Cotg. 

d. 

L.  Tang. 

L. 

Sin. 

d. 

' 

48°  30  . 

PP 

26 

35 

15 

M 

12 

ii 

.x 

2.6 

25 

.! 

1.5 

1.4 

.1 

1.2 

i.i 

.2 

5-2 

5-o 

.2 

3-o 

2.8 

.2 

2.4 

2.2 

•3 

7.8 

7-5 

3 

45 

4-2 

•3 

3-6 

3-3 

•4 

10.4 

IO.O 

•4 

6.0 

5-6 

•4 

4.8 

4-4 

13.0 

12.5 

7-5 

7.0 

.5 

6.0 

5-5 

.6 

15.6 

15-0 

.6 

9.0 

8.4 

.6 

7.2 

6-.  6 

•  7 

18.2 

I7-5 

•7 

10.5 

9.8 

•7 

8.4 

7-7 

.8 

20.8 

20.  o 

.8 

12.0 

II.  2 

.8 

9.6 

8-8 

0 

23.4           22.5 

•9 

13.5             12.6 

•9 

10.8              9.9 

41°  3D . 


' 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

30 

9.82  126 

9.94  68  1 

o.o5  319 

9-  87  446 

30 

3i 

9.82  i4i 

9.94  706 

25 
26 

o.o5  294 

9.87434 

12 

29 

32 

9.82  155 

9.94732 

o.o5  268 

9.87423 

28 

33 

9.82  169 

15 

9.94757 

25 
26 

o.o5  243 

9.87412 

27 

34 

9.82  1  84 

9.94  783 

o.o5  217 

9.87  4oi 

26 

35 

9.82  198 

9.94  808 

o.o5  192 

9.87  390 

25 

36 

9.82  212' 

14 

9.  94834 

o.o5  1  66 

9.87  378 

12 

24 

37 

9.82  226 

14 

9-9485 

9 

25 

o.o5  i 

4i 

9.87367 

II 

23 

38 

9.82  240 

9.94884 

o.o5  116 

9.87  356 

22 

39 

9.82  255 

15 

9.94  910 

o.o5  090 

9.87345 

II 

21 

40 

9.82  269 

9.94935 

26 

o.o5  065 

9.87334 

II 

20 

4i 

9.82  283 

14 

9.94961 

25 

o.o5  039 

9.87  322 

'9 

42 

9.82  297 

9.94  986 

26 

o.o5  oi4 

9.87  3n 

18 

43 

9 

.823n 

9.95  OI2 

o.o4  9 

88 

9.87  3oo 

ll 

15 

25 

12 

44 

9 

.82  326 

9.95  037 

o.o4  963 

9.87  288 

16 

45 

9 

.82  34o 

9.95  062 

o.o4  938 

9.87  277 

i5 

46 

9 

.82  354 

9.95  088 

o.o4  912 

9.87  266 

i4 

25 

II 

47 

9 

.82  368 

9.95  n3 

26 

0.04887 

9.87255 

i3 

48 

9 

.82  382 

9.95  i3 

) 

o.o4  861 

9.87  243 

12 

49 

9 

.82  396 

14 

9.95  1  64 

25 

26 

o.o4836 

9.87  232 

II 

I  I 

50 

9 

.82410 

9.95  190 

o.o4  810 

9.87  221 

10 

5i 

9 

.82424 

15 

9.95  2i5 

25 

o.o4  785 

9.87  209 

9 

52 

9 

.82439 

9-95  24< 

i 

26 

o.o4  760 

9.87   I98 

8 

53 

9 

.82453 

14 

9  .95  266 

o.o4  734 

9.87187 

11 

7 

*4 

25 

12 

54 

9 

.82467 

9.95  291 

26 

o.o4  709 

9.8 

7  J75 

6 

55 

9 

.82  48  1 

9.953i7 

o.o4  683 

9.87  1  64 

5 

56 

9 

.82495 

9.95  342 

26 

o.o4658 

9.87  i53 

12 

4 

57 

9 

.82  509 

9.95  368 

25 

o.o4  632 

9.87  i4i 

3 

58 

9 

.82  523 

9.95  393 

o.o4  607 

9.87  i3o 

2 

59 

9 

.82  537 

»4 

9.95  4x8 

25 

26 

o.o4  582 

9.87  119 

I 

60 

9 

.8255i 

14 

9.95  444 

o.o4556 

9.87  107 

0 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L,  Tang. 

L. 

Sin. 

d. 

' 

48°. 

PP 

26 

35 

'5 

M 

12 

ii 

.1 

2.6 

2.5 

i 

i.s 

1.4 

.i 

1.2 

i.i 

.2 

5-2 

S-O 

2 

3-O 

2.8 

.2 

2.  4 

2.2 

3 

7.8 

7-5 

•3 

4-5 

4-2 

•  3 

3-6 

3-3 

4 

10.4 

IO.O 

4 

6.0 

5.6 

•4 

4.8 

4-4 

•  5 

13-0 

12.5 

^ 

7-5 

7.0 

-5 

6.0 

5-5 

.6 

15-6 

15-0 

6 

9.0 

8.4 

.6 

7.2 

6.6 

•7 

18.2 

17-5 

7 

10.5 

9.8 

.7 

8.4 

7-7 

.8 

20.8 

20.  o 

8 

I2.O 

II.  2 

.8 

9.6 

8.8 

23.4          22.5 

9 

13.5            12.6 

•9 

10  8 

n3 


42°. 


! 

L.  Sin. 

d. 

L.  Tang1,     d. 

L.  Cotg. 

L.  Cos. 

d. 

0 

9.82  55i 

14 

*4 
*4 
M 

J4 

*4 
J4 
14 
14 

9.95  444 

25 
26 
25 
25 
26 

25 
26 
25 
25 
26 

25 
25 
26 

25 
26 
25 
25 
26 
25 
26 

25 
25 
26 
25 
25 
26 
25 
26 
25 
25 

0.04  556 

9.87107 

ii 
ii 

12 
II 

12 
II 
II 
12 
II 

60 

i 

2 

3 

4 
5 
6 

8 
9 

9.82  565 
9.82679 
9.82  593 

9.82  607 
9.82  621 
9.82635 

9.82  649 
9.82663 
9.82  677 

9.95  469 
9.95495 

9.95  52O 

9.95545 

9.9557i 
9.95  596 

9.95  622 
9-95  647 
9.95  672 

o.o4  53i 
o.o4  5o5 
o.o4  48o 

0.04455 
o.o4  429 
o.o4  4o4 

o.o4  378 
o.o4353 
o.o4  328 

9.87  096 
9.87  085 
9.87073 

9.87062 
9.87  o5o 
9.87  039 

9.87028 
9.87  016 
9.87  005 

59 
58 
57 

56 
55 
54 

53 

52 

5i 

10 

9 

.82  691 

J4 
14 
'4 
M 
J4 
14 
13 
M 
X4 

9.95  698 

o.o4  3o2 

9.86993 

50 

1  1 

12 

i3 

i4 
i5 
16 

r? 

18 

19 

9 

9 
9 

9 
9 
9 

9 
9 
9 

.82705 
.82719 
.82733 

.82747 
.82  761 
.82775 

.82  788 
.82  802 
.82  816 

9.95  723 
9.95  748 
9<95  774 

9.95  799 
9.95825 
9.95  85o 

9.95875 
9.95  901 
9.95  926 

o.o4  277 

0.04  252 

o.o4  226 

O.O4  2OI 

o.o4  175 
o.o4  150 

o.o4  125 
o.o4  099 
o.o4  074 

9.86  982 
9.86  970 
9.86959 

9.86  947 
9.86936 
9.86  924 

9.86  913 
9.86  902 
9.86  890 

12 
II 
12 
II 
12 
II 
II 
12 

49 

48 

47 
46 
45 
44 

43 

42 

4i 

20 

9 

.82  83o 

9-95952 

o.o4  o48 

9.86  879 

40 

21 
22 
23 

24 
25 
26 

27 
28 
29 

9 
9 
9 

9 
9 

9 

9 
9 
9 

.82  844 
.82858 
.82872 

.82885 
.82  899 
.82  913 

.82  927 
.82941 
.82955 

14 
*4 
13 
»4 
«4 
J4 
14 
M 

9-95977 
9.96  002 
9.96  028 

9.96  o53 
9.96  078 
9.96  io4 

9.96  129 
9.96155 
9.96  180 

o.o4  023 
o.o3  998 
o.o3  972 

o.o3  947 
o.o3  922 
o.o3  896 

o.o3  871 
o.o3845 
o.o3  820 

9.86  867 
9.86855 
9.86844 

9.86832 
9.86  821 
9.86  809 

9.86  798 
9.86  786 
9.86775 

12 

12 
II 
12 
II 
12 
II 

39 
38 
37 
36 
35 
34 

33 

32 

3i 

30 

9 

.82  968 

9.96  2o5 

o.o3  795 

9.86  763 

30 

L.  COS. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L. 

Sin. 

d. 

- 

47°  3D  . 

PP 

.2 

•3 

•4 
•5 
.6 

:J 

26 

25 

M 

13 

.2 

•3 

•4 

•5 
.6 

•7 
.8 

•9 

12 

ii 

2.6 

Si 

10.4 

13.0 

15.6 

18.2 

20.8 

23.4 

2-5 

50 

75 

10.0 

I2-5 
15-0 

i7-5 

20.0 

2 

•3 

•4 
-5 
.6 

1 

•9 

3 

4.2 

5-6 
7.0 
8.4 

9.8 

II.  2 

"•3 

2.6 

3.9 

5-2 

6-5 
7.8 

9.1 
10.4 
11.7 

1.2 

2-4 

3-6 

4.8 
6.0 
7.2 

b 

i.i 

2.  2 

3-3 
4-4 

1:1 

U 

u4 


42°  3O 


L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L. 

Cos. 

d. 

30 

9 

.82  968 

14 
14 
14 
13 
14 

*4 

*4 

*3 

J4 
M 

M 

13 
14 

*4 
»3 
*4 
14 
»3 
14 

9.96  2o5 

26 
25 
25 
26 

25 
25 
26 
25 
25 
26 

25 
26 
25 
25 
26 
25 
25 
26 
25 
25 
26 
25 
25 
26 
25 
25 
26 
25 
25 

26 

o.o3795 

9.86  763 

30 

3i 

32 

33 

34 
35 
36 

3? 

38 
39 

9 
9 
9 

9 
9 
9 

9 
9 

9 

.82  982 
.82  996 
.83oio 

.83023 
.83o37 
.83o5i 

.83065 
.83078 
.83  092 

9.96  23i 
9.96  256 
9.96  281 

9.96  307 
9.96  332 
9.96357 

9.96383 
9.96408 
9.96  433 

o.o3  769 
o.o3  744 
o.o3  719 

o.o3  693 
o.o3  668 
o.o3643 

o.o3  617 
o.o3  592 
o.o3  567 

9.86  762 
9.86  740 
9.86  728 

9.86  717 
9.86  705 
9.86694 

9.86  682 
9.86  670 
9.86  659 

12 
12 
II 
12 
II 
12 
12 
II 
12 

29 

28 
27 

26 

25 
24 

23 
22 
21 

40 

9 

.83  106 

9.96459 

o.o3  54i 

9.86  647 

20 

4i 

42 

43 

44 
45 
46 

4? 

48 

49 

9 
9 
9 

9 

9 
9 

9 
9 
9 

.83  120 
.83  i33 
.83  147 

.83  161 

.83  i74 
.83  188 

.83  202 

.832i5 
.83  229 

9.96484 
9.96  5io 
9.96535 

9.96  56o 
9.96  586 
9.96611 

9.96  636 
9.96662 
9.96  687 

o.o3  5i6 
o.o3  490 
o.o3465 

o.o3  44o 
o.o34i4 
o.o3  389 

o.o3  364 
o.o3338 
o.o3  3i3 

9.86635 
9.86624 
9.86  612 

9.86  600 
9.86589 
9.86577 

9.86565 
9.  86  554 
9.  86  542 

II 
12 
12 
II 
12 
12 
II 
12 

i9 

'7 
16 
i5 
i4 

i3 

12 
II 

50 

9 

.83  242 

9.96  712 

o.o3  288 

9.86  53o 

10 

5i 

52 

53 

54 
55 
56 

5? 
58 
59 

9.  83  256 
9.83  270 
9.83283 

9.  83  297 
9.83  3io 
9.83324 

9.83338 
9.8335i 
9.83  365 

H 
i3 
14 
»3 
H 
M 
13 
*4 
»3 

9.96  738 
9.96  763 
9.96  788 

9.96  8i4 
9.96  839 
9.96  864 

9.96  890 
9.96915 
9.96  940 

o.o3  262 
o.o3  237 

O.o3  212 

o.o3  186 
o.o3  161 
o.o3  i36 

o.o3  1  10 
o.o3o85 
o.o3  060 

9.865i8 
9.86  507 
9.86  495 

9.  86  483 
9.  86  472 
9.8646o 

9.  86  448 
9.  86  436 
9.  86  425 

II 

12 
12 
II 
12 
12 
12 
II 

9 

8 

7 

6 
5 

4 

3 

2 

60 

9 

.83378 

9.96  966 

o.o3  o34 

9.864i3 

0 

L.  Cos.       d. 

L.  Cotg.      d. 

L.  Tang. 

L. 

Sin. 

d. 

' 

47°. 

PP 

.1 

2 

•3 

•  4 

:l 

.7 

.8 
•9 

26 

25 

14 

13 

.1 

.2 

<3 
•4 

:i 

12                    II 

-2.6 

5-2 

7.8 

10.4 
13.0 
15-6 

18.2 

20.8 

2:5.4 

2.5                   .1 

5-O                            .2 

7-5                   -3 

10.0                          .4 

12.5          .5 

15.0                  .6 
17-5                    -7 

20.0                              8 

LI 

4.2 

5-6 
7-o 
8.4 

9.8 

II.  2 

12.6 

1:1 

3-9 

Is 

7.8 

9.1 
10.4 

11.7 

1.2                      I.I 
2.4                      2.2 

3-6              3-3 
4-8              4.4 

6'°         I'i 

7.2              6.6 

8.4              7.7 
9.6              8.8 

10.8               9.9 

i5 


43C 


/ 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

0 

9 

,83378 

9.96  966 

25 
25 
26 
25 
25 
26 

25 
25 
25 
26 

25 
25 
26 

25 

25 
26 
25 
25 
26 

25 

25 
26 
25 
25 
25 
26 
25 
25 
26 

25 

o.o3  o34 

9.864i3 

60 

2 

3 

4 
5 
6 

8 
9 

9.83  392 
9.834o5 
9.83419 

9.83432 
9.83446 
9.83459 

9.83473 
9.83486 
9.83  500 

*3 
*4 
i3 
14 
13 
H 
13 
J4 

9.96991 
9.97  016 
9.97  o4a 

9.97067 
9.97  092 
9.97  118 

9-97  '43 
9.97  168 
9.97  i93 

o.o3  009 
0.02  984 
0.02  958 

O.O2  933 
O.O2  908 
0.02  882 

0.02  857 

0.02  832 
O.O2  807 

9.86  4oi 
9.86  389 
9.86377 

9.86  366 
9.86  354 
9.86  342 

9.86  33o 
9.86  3i8 
9.86  3o6 

12 
12 
II 
12 
12 
12 
12 
12 

59 
58 
57 
56 

« 

53 

52 

5i 

10 

9 

.83  5i3 

J3 

9.97219 

O.O2  781 

9.86  295 

50 

1  1 

12 

i3 

i4 
i5 

16 

17 
18 

«9 

9.83  527 
9.83  54o 
9.83554 

9.83567 
9.  8358i 
9.83594 

9.83  608 
9.83  621 
9.83634 

i3 
N 

13 
H 
*3 
H 
13 
13 

9.97  244 
9.97  269 
9.97295 

9.97  320 

9.97345 
9.97371 

9.97  396 

9-97  42i 
9,97447 

0.02  756 
0.02  731 
0.02  705 

O.O2  680 
O.O2  65g 
O.O2  629 

0.02  6o4 

0.02  579 

0.02  553 

9.  86  283 
9.86  271 
9.86  259 

9.86  247 
9.86235 
9.86223 

9.86  211 
9.86  2OO 

9.86  188 

12 
12 
12 
12 
12 
12 
II 
12 
12 

49 

48 

47 
46 
45 
44 

43 

42 

4,i 

20 

9 

.83648 

9.97472 

O.02  528 

9.86  176 

40 

21 
22 
23 

a4 

25 

26 

27 
28 
29 

9 

9 
9 

9 
9 
9 

9 
9 
9 

.8366i 
.83674 
.83688 

.83  701 
.83  715 

.83  728 

.8374i 
.83  755 
.83  768 

»3 
*4 
»3 
r4 
*3 
»3 
H 

*3 

13 

9.97497 
9.97  523 
9-97548 

9.97573 
9.97  598 
9.97  624 

9.97649 
9.97  674 
9-97  7<>o 

0.02  5o3 

O.O2  477 
O.O2  452 

O.O2  427 
O.O2  402 
O.O2  376 

0.02  35i 

O.O2  326 

0.02  3oo 

9.86  1  64 
9.86  162 
9.86  i4o 

9.86  128 
9.86  116 
9.86  io4 

9.86  092 
9.86  080 
9.86  068 

12 
12 
12 
12 
12 
12 
12 
12 
12 

39 
38 

37 
36 

35 
34 
33 

32 

3r 

30 

9 

.83  781 

9-97  725 

O.02  275 

9.86  o56 

30 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L. 

Sin. 

d. 

' 

46°  30  . 

PP 

i 

2 

3 

4 
5 
6 

•  7 
.8 

•9 

26 

25 

14 

13 

.1 

.2 

•3 

•4 
•  5 
.6 

.7 

.8 

12 

ii 

2.6 

% 

10.4 

13.0 

15-6 

18.2 

20.8 

23.4 

2-5 

5-o 
7-5 

IO.O 

12.5 
15-0 

17-5 

20.  o 

.2 

•3 

•4 

:I 

•7 
.8 

•9 

a 

4.2 

5.6 
7.0 
8.4 

9.8 

II.  2 

12.6 

1.3 

2.6 

3-9 

5-2 

6.5 
7.8 

9.1 
10.4 

11.7 

1.2 

2-4 

3-6 

4.8 
6.0 
7-2 

8.4 
9.6 
10.8 

1.  1 

2.2 

3-3 

4-4 

II 

7-7 
8.8 

Q.Q 

n6 


43°  3O 


I 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

30 

9 

83  781 

9-97725 

25 
26 
25 
25 
25 
26 

25 
25 
26 

25 

25 
26 
25 
25 
25 
26 

25 
25 
26 
25 

25 
25 
26 

25 
25 
26 
25 
25 
25 
26 

O.O2  275 

9- 

86o56 

30 

3i 

32 

33 

34 
35 
36 

37 
38 
39 

9.  83  795 
9.  83  808 
9.83  821 

9.83834 
9.83  848 
9-83  861 

9.83  874 
9.83  887 
9-83  901 

13 

13 

13 
13 

13 

9.97750 
9.97  776 
9.97  801 

9.97  826 
9.97  85i 
9.97877 

9.97902 
9.97927 
9.97953 

0.02  250 
O.O2  224 
O.O2  199 

O.O2  174 
O.O2  iJcj 
O.O2  123 

O.O2  098 
O.O2  O73 

0.02  o47 

000  000  OOO 

86o44 
86o32 
86  020 

86  008 
85  996 
85  984 

85  972 
85  960 

85  948 

12 
12 
12 
12 
12 
12 
12 
12 

29 

28 
27 

26 

25 

24 

23 
22 
21 

40 

9 

83  9i4 

13 

9.97  978 

0.02  022 

9 

85936 

20 

4i 

42 

43 

44 
45 
46 

47 
48 

49 

9 
9 

9 

9 
9 
9 

9 
9 
9 

83  927 
83  940 
83  954 

83  967 
83  980 
83  993 

84  006 

84  020 
84o33 

13 
14 

13 
13 

9.98  oo3 
9.98  029 
9.98  o54 

9.98079 
9.98  io4 
9.98  i3o 

9.98  i55 
9.98  180 
9.98  206 

o.oi  997 
o.oi  971 
o.oi  946 

o.oi  921 
o.oi  896 
o.oi  870 

o.oi  845 
o.oi  820 
o.oi  794 

ooo  ooo  ooo 

85  924 
85  912 
85  900 

85888 
85  876 
85  864 

8585i 
85839 
85827 

12 
12 
12 
12 

12 

'3 
12 
12 
12 

12 
12 
12 
13 
12 
12 
12 
12 
12 
13 

'9 

18 
17 

16 
i5 
i4 
i3 

12 
I  I 

50 

9 

84  o46 

9.98  23i 

o.oi  769 

9 

85  8i5 

10 

9 

8 

7 

6 
5 
4 
3 

2 
I 

5i 

52 

53 

54 
55 
56 

57 
58 
59 

9.84059 
9.84  072 
9.84  o85 

9.84  098 
9.84  112 
9.84  125 

9.84  1  38 
9.84  1  5  1 
9.84  1  64 

13 
'3 

'3 

'3 
'3 

9.98  256 
9.98  281 
9.98  307 

9.98  332 
9.98357 
9.  98  383 

9.98408 
9.98433 
9.98  458 

o.oi  744 
o.oi  719 
o.oi  693 

o.oi  668 
o.oi  643 
o.oi  617 

o.oi  592 
o.oi  567 

O.OI  542 

9 
9 
9 

9 
9 

9 
9 
9 

858o3 
8579i 
85779 
85  766 
85754 
85  742 

8573o 

85  718 
85  706 

60 

9 

.84177 

9.  98  484 

o.oi  5i6 

9 

85693 

0 

L.  Cos.       d. 

L.  Cotg.      d. 

L.  Tang. 

L.  Sin. 

d. 

f 

46°. 

PP 

.1 

.2 

•3 

•4 
•5 
.6 

•7 
.8 

26 

25 

.2 

•3 
•4 

14 

13 

12 

2.6 

S-2 
7.8 

10.4 
13.0 

15-6 

18.2 

20.8 

23-4 

2-5 
5-o 
7-5 

10.0 

12.5 
15.0 

17-5 

20.  o 

1.4 

2.8 
4-2 

5.6 
7.0 
8.4 

9.8 

II.  2 

1.3 

2.6 

3-9 

5-2 
6-5 
7.8 

9.1 
10.4 

11.7 

.1 

.2 

•3 
•4 

•7 
.8 

1.2 

2-4 

3-6 

4.8 
6.0 

7-2 

8.4 
9.6 
10.8 

117 


44°. 


/ 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos.     d. 

0 

9 

84i?7 

13 

9.98  484 

o.oi  5i6 

9- 

85  693 

60 

! 

9 

84  190 

13 

9.9 

8  509 

25 

o.oi  491 

9 

8568i 

12 

59 

2 

9 

842o3 

9.98534 

o.oi  466 

9 

85  669 

58 

3 

9 

84216 

9.98  56o 

o.oi  44o 

9 

85  657 

57 

J3 

25 

4 

9 

84  229 

13 

9.9 

858s 

25 

o.oi  4i5 

9 

85  645 

13 

56 

5 

9 

84242 

9.98  610 

o.oi  390 

9 

85  632 

55 

6 

9 

.84255 

M 

9.  98  635 

25 
26 

o.oi  365 

9 

85  620 

12 

54 

7 

9.84  269 

1-3 

9.98  661 

25 

O.OI   339 

9 

85  608 

12 

53 

8 

9 

.84282 

9.98  686 

o.oi  3i4 

9 

85  596 

52 

9 

9 

.84295 

J3 

9.98  711 

25 

_fC 

o.oi  289 

9 

85583 

J3 

5i 

10 

9 

.843o8 

9.98  737 

o.oi  263 

9 

8557i 

50 

ii 

9 

.84  3ai 

13 

9.98  762 

25 

o.oi  238 

9 

85  559 

49 

12 

9 

.84334 

9.98787 

O.OI   2l3 

9 

85  547 

48 

i3 

9 

.84347 

13 

9.98  812 

25 
26 

o.oi  188 

9 

85534 

I3 

12 

47 

i4 

9 

.8436o 

13 

9.98838 

o.oi  162 

9 

85  522 

46 

i5 

9 

.84373 

9.9 

8  863 

25 

o.oi  137 

9 

85  5io 

45 

16 

9 

.84  385 

9.98  888 

25 

O.OI    112 

9 

.85497 

*3 

44 

i? 

9 

.84398 

*3 
13 

9.98  913 

25 
26 

o.oi  087 

9 

.85485 

12 

43 

18 

9 

.844ii 

9.98939 

o.oi  06  i 

9 

.85473 

42 

'9 

9 

.84424 

9.98  964 

25 

o.oi  o36 

9 

.85  46o 

J3 

4i 

20 

9 

.84437 

9.98989 

25 
„/: 

O.OI   OI  I 

9 

85448 

40 

21 

9 

.8445o 

I3 

9.99015 

25 

o.oo  985 

9 

.85436 

39 

22 

9 

.84463 

9.99  o4o 

o.oo  960 

9 

.85423 

38 

23 

9 

.84476 

9.99  o65 

25 

o.oo  935 

9 

.854ii 

37 

X3 

25 

12 

24 
25 

9 
9 

.84489 
.84  5o2 

13 

9.99090 
9.99  1  1  6 

26 

o  .  oo  910 

0.00884 

9 
9 

.85  399 
.85386 

13 

36 
35 

26 

9 

.845i5 

9.99  i4i 

25 

o.oo  859 

9 

.85374 

34 

27 

9 

.84528 

*3 

9.99  1  66 

25 

o.oo  834 

9 

.8536i 

*3 

33 

28 

9 

.8454o 

9.99  191 

o.oo  809 

9 

.85349 

32 

29 

9 

.84553 

'3 

9.99  217 

o.oo  783 

9 

.85  337 

12 

3f 

30 

9 

.84566 

9.99  242 

25 

o.oo  758 

9 

.85  324 

J3 

30 

L.  Cos.      d. 

L.  Cotgr. 

d. 

L.  Tang. 

L.  Sin.     d. 

/ 

45°  30  . 

PP 

26 

«5 

M 

13 

12 

.1 

2.6 

2-5 

.1 

1-4 

1.3 

.1 

1.2 

.2 

5-2 

5-o 

.2 

2.8 

2.6 

.2 

2.4 

•3 

7.8 

7-5 

•  3 

4.2 

3-9 

•3 

3-6 

•4 

10.4 

10.0 

•4 

5-6 

5-2 

•4 

4-8 

•5 

13.0 

12.5 

•5 

7.0 

6.5 

.5 

6.0 

.6 

15-6 

15.0 

.6 

8.4 

7.8 

.6 

7.2 

•7 

18.2 

17.5 

•  7 

9-8 

9-1 

•7 

8.4 

.8 

20.8 

20.0 

.8 

II.  2 

10.4 

.8 

9.6 

•9 

23.4             22.5 

.9            12.6 

"7 

.9            10.8 

118 


44°  3O 


/ 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

30 

9 

.84  566 

13 

*3 
i3 
13 

12 
13 
»3 
13 

r3 

12 

»3 
13 

'3 

12 
13 
»3 
»3 
f2 

*3 

'3 
'3 

12 
13 

*3 

12 

'3 

13 

12 

'3 

9.99  242 

25 

26 

25 
25 
25 
26 

25 
25 

25 
26 

25 
25 
25 
26 

25 
25 
26 
25 
25 
25 
26 
25 
35 
25 
26 
25 
25 
25 
26 
25 

o.oo  758 

9 

.85  324 

30 

3  1 

32 

33 

34 
35 
36 

3? 
38 
39 

9 
9 
9 

9 
9 
9 

9 

9 
9 

.84579 
.84  592 
.84  605 

.846i8 
.8463o 
.84643 

.84656 
.84669 
.84682 

9.99  267 
9  99  293 
9.99  3i8 

9.99  343 
9.99  368 
9.99394 

9.99419 
9.99  444 
9.99469 

o.oo  733 
o.oo  707 
o.oo  682 

o.oo  657 
o.oo  632 
o.oo  606 

o.oo  58  1 
o.oo  556 
o.oo  53i 

9 
9 
9 

9 
9 

9 

9 
9 
9 

.853i2 
.85  299 
.85  287 

.85274 
.85  262 
.85  250 

.85  237  - 
.85  225' 

.85  212 

*3 
12 

13 
12 
12 
»3 
-12 
»3 

29 

28 
27 

26 

25 
24 

23 
22 
21 

40 

9 

.84  694 

9.99495 

o.oo  5o5 

9 

.85  200 

20 

4i 

42 

43 

44 
45 
46 

4? 
48 

49 

9.84  707 
9.84  720 
9.84733 

9-84745 
9.  84  758 
9-84  771 

9.84784 
9.84  796 
9.84  809 

9.99  52O 

9.99  545 

9.99  570 

9.99  596 
9.99  621 
9.99  646 

9.99672 
9.99697 
9.99722 

o.oo  48o 
o.oo  455 
o.oo  43o 

o  .  oo  4o4 
o.oo  379 
o.oo  354 

O.OO  328 

o.oo  3o3 
o.oo  278 

9 
9 
9 

9 
9 
9 

9 
9 
9 

.85  187 
.85175 
.85  162 

.85  150 
.85  i37 

.85  125 

.85  112 
.85  100 
.85087 

13 

12 
13 
12 

13 
12 
13 
12 

*3 

;89 

17 

16 
i5 

i4 

i3 

12 
I  I 

50 

9 

.84  822 

9.99  747 

o.oo  253 

9 

.85  074 

10 

5i 

52 

53 

54 
55 
56 

5? 

58 

59 

9 
9 
9 

9 
9 

9 

9 
9 

9 

.84835 

.84847 
.8486o 

.84873 
.84885 
.84898 

.84911 
.84923 
.84936 

9-99773 
9.99798 
9.99  823 

9.99848 
9.99874 
9.99899 

9.99924 
9.99949 
9-99975 

o.oo  227 

O.OO  2O2 

o.oo  177 

O.OO  I  52 

o.oo  126 

O.OO  IOI 

o.oo  076 
o.oo  o5i 

O.OO  O25 

9 
9 
9 

9 
9 
9 

9 

9 
9 

.85  062 
.85  049 
.85o37 

.85o24 
.85  012 
.84  999 

.84986 
.84974 
.84961 

»3 

12 

13 
12 

»3 
13 
12 

13 

9 

8 

7 
6 
5 

4 

3 

2 
I 

60 

9 

.84949 

r3 

0.00  OOO 

0.00  OOO 

9 

.84949 

0 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L.  Sin. 

d. 

' 

45°. 

PP 

.2 

•3 
•4 

:! 
:l 

•9 

26 

25 

.  i 

.2 

-3 
•4 

:I 

•9 

14 

13 

.1 

.2 

•3 

•4 
•  5 
.6 

•7 
.8 

•9 

12 

2.6 

5-2 

7.8 

10.4 

13.0 

15-6 

18.2 

20.8 

23-4 

2.5 

5-o 
7-5 

IO.O 

12.5 
15.0 

17-5 
20.  o 

22-5 

a 

4.2 

5.6 
7.0 
8.4 

9.8 

II.  2 

12.6 

i-3 

2.6 

3-9 

5-2 

6-5 

7.8 

9.1 
10.4 
11.7 

1.2 
2-4 

3-6 

4.8 
6.0 
7-2 

8.4 
9.6 

10.8 

119 


TABLE   III 
FIVE-PLACE    LOGARITHMS 

OF  THE 

SINE    AND    TANGENT    OF 
SMALL    ANGLES 

THE  SINE  AND  TANGENT  TO  EVERY  SECOND  FROM  O°  TO  8'  J  TO  EVERY 
TEN  SECONDS  FROM  O°  TO  2°. 

THE  COSINE  AND  COTANGENT  TO  EVERY  SECOND  FROM  90°  TO  89° 
52'  ;  TO  EVERY  TEN  SECONDS  FROM  90°  TO  88°. 


0°. 


FUNCTIONS   OF   SMALL   ANGLES 

LOGARITHMIC  SINE  AND  TANGENT. 


0" 

1" 

2" 

3" 

4" 

5" 

6" 

7" 

8" 

9" 

10" 

0  o 

10 
20 

5-  68557 
98660 

68557 
72697 

*°°779 

98660 
76476 
$02800 

#16270 
79952 

*0473Q 

,28763 
83170 
1*06579 

#38454 
86167 
#08351 

#46373 
88969 

#IO°55 

*53o°7 
91602 
,11694 

#58866 
94085 
#13273 

#63982 
96433 
**4797 

*68557 
98660 
^16270 

50 
40 
30 

30 

40 

5° 

6.  16270 
28763 
38454 

29836 
39315 

19072 
30882 
40158 

20409 

3i904 
40985 

21705 
32903 
41797 

22964 
33879 
42594 

24188 
34833 
43376 

25378 
35767 
44145 

26536 
36682 
44900 

27664 

37577 
45643 

28763 
38454 
46373 

20 
IO 

o  59 

t  o 

IO 

20 

6.46373 
6.  5  3067 
8866 

7090 
3683 
9406 

7797 
4291 

9939 

8492 
4890 
#0465 

9»75 
548i 
#0985 

ft9 
6064 

**499 

•ss 

*2007 

*"6s 
7207 
*2509 

*'8f 
7767 
*3°o6 

*2442 

8320 
*3496 

*3°67 
8866 
*39g2 

50 
40 
3° 

30 
40 

50 

6.63982 

*  8F 
6.72697 

4462 
8990 
3090 

4936 
9418 
3479 

5406 
9841 
3865 

5870 
#0261 
4248 

633° 
^0676 
4627 

678! 

*io88 
5003 

7235 
*i496 
5376 

7680 
*i90Q 
5746 

8121 
*23°o 
6112 

8557 
#2697 
6476 

20 

10 

o  58 

2  o 

10 
20 

6476 
9952 
6.83170 

6836 

*028§ 

3479 

7»93 

*o6i§ 
3786 

7548 
*°943 
4091 

7900 
*i268 
4394 

8248 

*i59' 
4694 

8595 
*i9" 

4993 

8938 

*2230 
5289 

9278 

*2545 
5584 

9616 
+2859 
5876 

9952 
*3I7° 
6167 

50 
40 
30 

30 

40 

So 

6167 
896g 
6.9  1602 

6455 
9240 
1857 

6742 
9509 

2110 

7027 
9776 
2362 

73io 

#0042 
2612 

759i 

#0306 
2861 

7870 
*o568 
3109 

8l47 
*0829 

3355 

8423 
*io88 

3599 

8697 
**346 
3843 

8969 
^1602 
4085 

20 
IO 

o  57 

3  o 

10 
20 

4085 

6433 
8660 

4325 
6661 
8877 

4565 

6888 
9°93 

4803 
7113 
93°7 

5039 
7338 
9520 

5275 
7561 

9733 

55°9 
7783 
9944 

5742 
8004 
*°i55 

5973 
8224 
#0364 

6204 
8443 
*0572 

6433 
8660 

*0779 

50 
40 
30 

3° 
40 
50 

7.00779 
2800 
4730 

0986 
2997 
49'9 

1191 

3193 
5106 

1395 
3388 
5293 

1599 
3582 
5479 

1801 
3776 
5664 

2003 
3968 
5849 

2203 
4160 
6032 

2403 
4351 
6215 

2602 
454i 
6397 

2800 
473° 
6579 

20 

IO 

o  58 

4  o 

10 

20 

6579 
8351 
7.  1  0055 

6759 
8525 

O222 

s 

7118 
8870 
0553 

7296 
9041 
0718 

7474 
9211 
0882 

7651 
9381 
1046 

7827 
955i 
1209 

8003 
9719 
1371 

8177 
9887 
1533 

8351 
*°°55 
1694 

50 
4° 
30 

3° 
40 
5° 

1694 
3273 

4797 

1854 
3428 

4947 

2014 
3582 
5096 

2174 
3736 
5244 

2333 
3889 

5392 

2491 
4042 
5540 

2648 
4194 
5687 

2805 
4346 
5833 

2962 
4497 
5979 

3»8 
4647 
6125 

3273 
4797 
6270 

20 
IO 

o  55 

5  o 

IO 
20 

7.16270 
7694 
9072 

6414 

7834 
9208 

6558 

7973 
9343 

6702 
8112 
9478 

6845 
8250 
9612 

6987 
8389 
9746 

7130 
8526 
9879 

7271 

8663 

#OOI2 

7413 
8800 
*oi45 

7553 
8937 
#0277 

7694 
9072 
#0409 

50 
40 

30 

30 

4° 

5" 

7.20409 
1705 
2964 

0540 
1833 
3088 

°67i 
1960 
3212 

0802 
2087 
3335 

0932 
2213 
3458 

1062 
2339 
358o 

1191 
2465 
3702 

I320 
2590 
3824 

1449 
2715 
3946 

1577 
2840 
4067 

i705 
2964 
4188 

20 
IO 

o  54 

6  " 

10 

20 

4188 
5378 
6536 

4308 

5495 
6650 

4428 
5612 
6764 

4548 
5728 
6877 

4668 
6991 

4787 
596i 
7104 

4906 
6076 
7216 

5024 
6192 
7329 

SH2 
6307 

744  > 

5260 
6421 
7552 

5378 
6536 
7664 

50 
40 
3° 

3° 
40 
So 

7  o 

IO 

20 

i 

7775 
8872 
9942 

7886 
8980 
*°°47 

7997 
9088 
#0152 

8107 
9196 
*0257 

8217 
9303 
*0362 

8327 
9410 

+0467 

8437 
9517 

*°571 

8546 
9623 
*o675 

8655 
9730 
*°779 

8763 
9836 
#0882 

20 
IO 

o  53 

7.30882 
1904 
2903 

0986 
2005 
3001 

1089 
2106 
3100 

1191 
2206 
3*98 

1294 
2306 
3296 

1396 
2406 
3393 

1498 
2506 

349  » 

1600 
2606 
3588 

1702 
2705 
3685 

1803 
2804 
3782 

1904 
2903 
3879 

50 
40 
30 

3° 
40 

50 

•M 

3879 
4833 
5767 

««M^^M» 

10" 

3975 

•—  «• 

4071 
5022 
5952 

4167 
5"6 
6044 

m^—m—i 

4263 
5209 
6i35 

—  «^— 

4359 
5303 
6227 

4454 
5396 
6318 

«__« 

4549 
5489 
6409 

4644 
5582 
6500 

4739 
5675 
6591 

—  — 

4833 
5767 
6682 

—  — 

20 
IO 

o  52 

•^•— 

9" 

8" 

7" 

6" 

5" 

4" 

3" 

2" 

1" 

0" 

LOGARITHMIC  COSINE  AND  COTANGENT. 


89C 


FUNCTIONS    OF    SMALL    ANGLES. 


' 

L.  Sin.   L.  Tang. 

L.  Sin. 

L.  Tang. 

0   o 

IO 

20 

3o 
4o 
5o 

5.68557 
5.98  660 

6.  16  270 
6.28  763 
6.38454 

5.68557 
5.98  660 
6.  16  270 
6.28763 

6.38454 

o  60 

5o 
4o 
3o 
20 

10 

73o 
4o 
5o 

7.  33  879 
7.34833 
7.35767 

7.  33  879 
7.34833 
7.35767 

3o 
20 

IO 

8  o 

IO 
20 

3o 
4o 
5o 

7.36  682 
7.37577 
7.38454 

7.393i4 
7.40  1  58 
7.40985 

7.36682 
7.37577 
7-38455 

7-393i5 
7.40  i58 
7.40985 

o  52 

5o 
4o 
3o 
20 

IO 

1   o 

IO 

20 
3o 
4o 
5o 

6.46  373 
6.53o67 
6.58866 
6.  63  982 
6.68557 
6.72  697 

6.46  373 
6.53o67 
6.58866 
6.  63  982 
6.68557 
6.72  697 

o  59 

5o 
4o 
3o 
20 

IO 

9  o 

10 

20 
3o 
4o 
5o 

7-4i  797 
7.42  594 
7.43376 

7.44  i4s 
7.44  900 
7.45643 

7-4i  797 
7.42  594 
7.43376 

7.44i45 
7-44  900 
7.45643 

o  51 

5o 
4o 
3o 
20 

IO 

2  o 

IO 
20 

3o 
4o 
5o 

6.76476 
6.79952 
6.83  170 
6.86  167 
6.88  969 
6.91  602 

6.76476 
6.79  952 
6.83  170 

6.86  167 
6.88  969 
6.91  602 

o  58 

5o 
4o 
3o 
20 

10 

10  o 

IO 

20 
3o 
4o 
5o 

7.46373 
7.47  090 
7.47797 
7.48491 
7.49  175 
7.49849 

7.46373 
7.47091 
7.47797 
7.48492 
7.49  176 
7.49  849 

o  50 

5o 
4o 
3o 

20 
10 

3  o 

10 
20 

3o 

4o 
5o 

6.94085 
6.96433 
6.98  660 
7.00779 
7.02  800 
7.04  73o 

6.94085 
6.96433 
6.98  660 

7.00779 
7.02  800 
7.04  73o 

o  57 

5o 
4o 
3o 

20 
10 

11  o 

IO 

20 
3o 
4o 
5o 

7-5o  5i2 
7-5i  165 
7.5i  808 

7.52  442 
7.53067 
7.53683 

7«5o  5i2 
7.5i  i65 
7.  5  1  809 

7.52443 
7.53  067 
7.53683 

o  49 

5o 

4o 
3o 

20 
10 

4  o 

10 

20 
3o 
4o 

5o 

7.o6579 
7.o835i 
7.10  o5s 

7.11  694 
7.i3273 
7.14  797 

7.06  579 
7.  08  352 
7.10  o55 

7.11  694 
7.i3273 
7.14797 

o  56 

5o 
4o 
3o 

20 
10 

12  o 

10 
20 

3o 

4o 
5o 

7.54  291 
7.54890 
7.5548i 

7.56  o64 
7.56639 
7.57  206 

7.54  291 
7.54  890 
7.5548i 

7.56o64 
7.56639 
7.57207 

o  48 

5o 
4o 
3o 
20 

IO 

5  o 

IO 
20 

3o 
4o 
5o 

7.  16  270 

7-  1?694 
7.  19  072 

7.20  409 
7.21  7o5 
7.22  964 

7.  16  270 
7.17694 
7.19073 

7.20  409 
7.21  705 
7.22  964 

o  55 

5o 
4o 
3o 

20 
10 

13  o 

IO 

20 
3o 
4o 
5o 

7.57767 
7.58  320 
7-58866 
7.59  4o6 
7.59939 
7.60  465 

7.57767 
7.58  320 
7.58867 
7.59  4o6 
7.59939 
7.60466 

o  47 

5o 
4o 
3o 
20 

10 

6  o 

10 

20 
3o 
4o 
5o 

7.24  188 
7.25  378 
7.26  536 

7.27  664 
7.28763 
7.29  836 

7.24  188 
7.25  378 
7.26536 
7.27  664 
7.28764 
7.29  836 

o  54 

5o 

4o 
3o 

20 
10 

14  o 

10 
20 

3o 
4o 
5o 

7.60985 
7.61  499 
7.62  007 

7.62  5og 
7.63  006 
7.63496 

7.60  986 
7.61  500 
7.62  008 
7.62  5io 
7.  63  006 
7-63  497 

o  46 

5o 
4o 
3o 
20 

IO 

7  o 

IO 

20 
3o 

7.3o  882 
7.3i  904 
7.32  go3 

7.  33  879 

7.30882 
7.  3  1  904 
7  .  32  go3 

7.  33  879 

o  53 

5o 

4o 
3o52 

15  o 

7.63  982 

7.63  982 

o  45 

L.  Cos. 

L.  Cotg. 

a    i 

L.  Cos. 

L.  Cotg. 

n    , 

123 


89°. 


FUNCTIONS    OF    SMALL    ANGLES. 
0°. 


r      rt 

L.  Sin. 

L.  Tang. 

,     „ 

L.  Sin. 

L.  Tang. 

15  o 

10 

20 
3o 
4o 
5o 

7.63  9»2 
7-64461 
7.64936 
7.654o6 
7.65  870 
7.6633o 

7.63  982 
7.64462 
7.64937 
7.65  4o6 
7.65871 
7.6633o 

o  45 

5o 
4o 
3o 
20 

10 

22  3o 
4o 
5o 

7  .81  591 
7.81  911 
7.82  229 

7.81  59i 
7.81  912 
7.82  23o 

3o 

20 
IO 

23  o 

IO 

20 
3o 
4o 
5o 

7.82  545 
7.82  859 
7.  83  i7o 

7.83479 
7.83786 
7.84  091 

7.82  546 
7.82860 
7.83  171 

7.8348o 
7.  83  787 
7  .84  092 

o  37 

5o 
4o 
3o 

20 
10 

16  o 

10 
20 

3o 
4o 
5o 

7.66  784 
7.67235 
7.67  680 

7.68  121 

7.68557 
7.68  989 

7.66785 
7.67235 
7.67680 

7.68   121 

7.68558 
7.68  990 

o  44 

5o 
4o 
3o 
20 

IO 

24  o 

IO 

20 
3o 
4o 
5o 

7.84  393 
7-84694 
7.84  992 
7.85  289 
7.85583 
7.85876 

7-84394 
7.84695 
7.84993 
7.85  290 
7.85  584 
7.  85  877 

o  36 

5o 
4o 
3o 
20 

IO 

17  o 

IO 
20 

3o 

4o 
5o 

7-69417 
7.69  84i 
7.  70  261 

7.70676 
7.71  088 
7-71  496 

7.69418 
7.69  842 
7.70  261 
7.70677 
7.71  088 
7-71  496 

o  43 

5o 
4o 
3o 
20 

10 

25  o 

IO 

20 
3o 
4o 
5o 

7.86  166 
7.  86  455 
7  86  741 

7.87  026 
7.87  309 
7.87590 

7.86  167 
7.86456 
7.86743 
7.87027 
7.87310 
7.8759i 

o  35 

5o 
4o 
3o 

20 
IO 

18  o 

10 
20 

3o 
4o 
5o 

7.71  900 
7.72  3oo 
7.72697 
7.73090 

7.73479 
7-73865 

7.71  900 
7,72  3oi 
7.72697 
7.73  090 
7.7348o 
7.73866 

o  42 

5o 

4o 
3o 

20 
10 

26  o 

10 

20 
3o 

4o 
5o 

7.87870 
7.88  147 
7-88423 

7.88697 
7.88  969 

7.89  24o 

7.8787i 
7.88  i48 
7.88424 
7.88  698 
7.88  970 
7.89  241 

o  34 

5o 
4o 
3o 
20 

IO 

19  o 

10 

20 

3o 

4o 
5o 

7.74248 
7-74627 
7.75  oo3 
7.75376 
7.75745 

7.76  112 

7.74248 
7.74628 
7.75  oo4 
7.75377 
7.75746 
•7.76  n3 

o  41 

5o 
4o 
3o 

20 
10 

27  o 

IO 

20 

3o 
4o 
5o 

7.89  509 

7.89776 
7.90  o4  1 
7.90  3o5 
7.90  568 
7.90  829 

7.89  5io 
7.89777 
7.90043 
7.90307 
7.90  569 
7.90  83o 

o  33 

5o 
4o 
3o 

20 
10 

20  o 

IO 
20 

3o 
4o 
5o 

7.76475 

7.  76  836 
7-77  193 
7-77548 
7.77899 
7.78248 

7.76476 
7.76837 

7«77  i94 
7.77549 
7.77900 
7.78  249 

o  40 

5o 
4o 
3o 

20 
IO 

28  o 

IO 
20 

3o 
4o 
5o 

7.91  088 
7.91  346 
7.91  602 
7.9i857 
•7.92  1  10 
7.92  362 

7.91  o89 

7.91347 
7.9,1  6o3 

7.91  858 
7.92  in 
7.92  363 

o  32 

5o 
4o 
3o 
20 

10 

21  o 

10 

20 
3o 
4o 
5o 

7.78594 
7.78938 
7.79278 
7.79616 
7.79952 
7.80284 

7.78595 
7.78938 
7.79279 

7.79617 
7.79952 
7.80285 

o  39 

5o 
4o 
3o 
20 

10 

29  o 

IO 
20 

3o 
4o 
5o 

7.92  612 
7.92  861 
7.93  108 

7.93354 
7.93599 
7.93  842 

7.92  6i3 
7.92  862 
7.93  no 

7.93  356 
7.93  601 
7-93844 

o  31 

5o 
4o 
3o 

20 
10 

22  o 

IO 
20 

3o 

7.80615 

7.80  942 
7.81  268 
.81  591 

7.8o6i5 
7.80943 
7.81  269 
7.81  591 

o  38 

5o 

4o 

3o37 

30  o 

7.94  o84 

7.94  086 

o  30 

L.  Cos.    L.  Cotg. 

"    ' 

L.  Cos.  i  L.  Cotg. 

//    / 

89°. 


FUNCTIONS    OP    SMALL   ANGLES. 
0°. 


,    " 

L.  Sin. 

L.  Tang. 

,       „ 

L.  Sin. 

L.  Tang. 

30  o 

10 
20 

7.94  o84 
7.94325 
7.94564 

7.94086 
7.94  326 
7.94  566 

o  30 

5o 
4o 

37  3o 
4o 
5o 

8.o3775 
8.o3  967 
8.o4  1  59 

8.o3777 
8.o3  97o 
8.o4  162 

3o 
20 

10 

3o 
4o 
5o 

7.94  802 
7.96  o39 
7.96  274 

7.94804 
7.96  o4o 
7.95  276 

3o 
20 
10 

38  o 

IO 

20 

8.o435o 
8.o4  54o 
8.o4  729 

8.o4353 
8.o4543 
8.o4732 

o  22 

5o 
4o 

31  o 

10 
20 

7.96  5o8 
7.9574i 
7.95  973 

7.95  5io 
7.95743 
7.95974 

o  29 

5o 
4o 

3o 
4o 
5o 

8.o4  918 
8.o5  io5 
8.o5  292 

8.o4  921 
8.o5  108 
8.o5  295 

3o 
20 

IO 

3o 
4o 
5o 

7.96  2o3 
7.96  432 
7.96  660 

7.96  205 
7.96434 
7.96  662 

3o 

20 
10 

39  o 

IO 
20 

8.05478 
8.o5663 
8.o5848 

8.o548i 
8.o5666 
8.o585i 

o  21 

5o 

4o 

32  o 

10 

20 

7.96887 
7.97  n3 
7.97337 

7.96889 

7-97  n4 
7.97339 

o  28 

5o 

4o 

3o 
4o 
5o 

8.o6o3i 
8.06214 
8.06  396 

8.o6o34 
8.06  217 
3.o6  399 

3o 
20 

10 

3o 
4o 
5o 

7.97660 

7-97  782 
7.98  oo3 

7.97662 

7.97  784 
7.98  oo5 

3o 

20 
10 

40  o 

IO 

20 

8.06678 
8.o6758 
8.o6938 

8.o658i 
8.06  761 
8.06  94  1 

o  20 

5o 
4o 

33  o 

10 

20 

7.98  223 
7.98442 
7.98  660 

7.98  225 

7.98444 
7.98  662 

o  27 

5o 
4o 

3o 
4o 
5o 

8.07  117 
8.07  296 
8.07473 

8.07  I2O 
8.07  298 

8.o7476 

3o 

20 
IO 

3o 
4o 
5o 

7.98  876 
7.99  092 

7.99  3o6 

7.98  878 
7.99  094 

7.99  3o8 

3o 
20 

IO 

41  o 

IO 
20 

8.07  650 
8.07  826 
8.08  002 

8.  07  653 
8.07  829 
8.08  005 

o  19 

5o 
4o 

34  o 

10 
20 

7.99  620 
7.99  732 
7.99943 

7.99  522 
7.99  734 

7.99  946 

o  26 

5o 
4o 

3o 
4o 
5o 

8.08  176 
8.o835o 
8.08  624 

8.08  1  80 
8.08  354 
8.08  627 

3o 
20 

IO 

3o 
4o 
5o 

8.00  1  54 
8.00  363 
8.00  571 

8.00  1  56 
8.oo365 
8.00  574 

3o 
20 

10 

42  o 

IO 

20 

8.08696 
8.08868 
8.09  o4o 

8  .08  700 
8.08  872 
8.09043 

o  18 

5o 
4o 

35  o 

10 
20 

8.00  779 
8.00  985 
8.01  190 

8.00  781 
8.00987 
8.01  193 

o  25 

5o 
4o 

3o 
4o 
5o 

8.09  2IO 

8.09  38o 
8.09  650 

8.09  214 
8.09  384 
8.  09  553 

3o 
20 
10 

3o 
4o 
5o 

8.01  395 
8.01  598 
8.01  801 

8.01  397 
8.01  600 
8.01  8o3 

3o 

20 
10 

43  o 

IO 

20 

8.09  718 
8.09  886 
8.ioo54 

8.09  722 
8.09  890 
8.  10  067 

o  17 

5o 
4o 

36  o 

10 

20 

8.02  OO2 
8.02  203 

8.  02  402 

8.  02  oo4 
8.  02  2o5 
8.02  405 

o  24 

5o 
4o 

3o 
4o 
5o 

8.  10  220 
8.io386 
8.10  552 

8.  10  224 
8.10  390 
8.io555 

3o 

20 
IO 

3o 
4o 
5o 

8.  02  601 
8.  02  799 
8.02  996 

8.02  6o4 
8.02  801 
8.02  998 

3o 

20 
10 

44  o 

10 
20 

8.10  717 
8.10  881 
8  .  1  1  o44 

8.  10  720 
8.io884 
8.  1  1  o48 

o  16 

5o 

4o 

37  o 

10 

20 

8.o3  192 
8.o3  387 
8.o3  58i 

8.o3  194 
8.o339o 
8.o3  584 

o  23 

5o 
4o 

3o 

4o 
5o 

8.  1  1  207 
8.  1  1  370 
8.  ii  53i 

8.  I  I  211 

8.  ii  373 
8.  ii  535 

3o 

20 
IO 

3o 

8.03775 

8.o3777 

3o22 

45  o 

8.  1  1  693 

8.  ii  696 

3     15 

L.  Cos. 

L.  Cotg. 

"    ' 

L.  Cos. 

L.  Cotg. 

"    ' 

125 


89°. 


FUNCTIONS    OP    SMALL    ANGLES. 


/      // 

L.  Sin. 

L.  Tang. 

/        tr 

L.  Sin.    L.Tang. 

45  o 

10 

20 
3o 
4o 
5o 

8.11  693 
8.  ii  853 
8.i2oi3 
8.  12  172 
8.i233i 
8.12489 

8.  n  696 
8.  ii  857 
8.12  017 
8.12  176 
8.12335 
8.12493 

o  15 

5o 
4o 
3o 

20 
10 

52  3o 
4o 
5o 

8.18  387 
8.18  524 
8.18662 

8.18  392 
8.18  53o 
8.18667 

3o 

20 
IO 

53  o 

10 
20 

3o 
4o 
5o 

8.18  798 
8.18  935 
8.19  071 
8.19  206 
8.i934i 
8.19476 

8.18  8o4 
8.18  940 
8.19076 

8.  19  212 

8.19347 
8.19481 

o    7 

5o 
4o 
3o 

20 
IO 

46  o 
10 

20 
3o 
4o 
5o 

8.12  647 
8.12  8o4 
8.12  961 

8.i3  117 
8.i3  272 
8.13427 

8.12  65i 
8.12  808 
8.12  965 

8.i3  121 
8.i3276 
8.i343i 

o  14 

5o 
4o 
3o 
20 

IO 

54  o 

10 
20 

3o 
4o 
5o 

8.19610 

8.19744 
8.19877 

8.20  OIO 

8.20  i43 
8.  20  275 

8.19616 
8.19  749 

8.19  883 
8.  20  016 
8.  20  149 
8.20  281 

o    6 

5o 
4o 
3o 

20 
IO 

47  o 

10 

20 
3o 
4o 
5o 

8.i3  58i 
8.i3735 
8.13888 
8.i4o4i 
8.i4  193 
8.i4344 

8.i3  585 
8.i3  739 
8.i3  892 
8.14045 
8.i4  197 
8.i4348 

o  13 

5o 
4o 
3o 
20 

IO 

55  o 

IO 
20 

3o 
4o 
5o 

8.20  407 
8.20538 
8.20  669 

8.20  800 
8.20  930 
8.21  060 

8.20  4i3 
8.20  544 
8.20675 

8.20806 
8.20936 
8.21  066 

o    5 

5o 
4o 
3o 

20 
10 

48  o 

10 

20 
3o 
4o 
5o 

8.14496 
8.i4646 
8.i4  796 
8.i4945 
8.  1  5  094 
8.i5243 

8.i4  500 
8.i465o 
8.i48oo 

8.  i4  950 
8.  1  5  099 
8.i5  247 

o  12 

5o 
4o 
3o 

20 
IO 

56  o 

IO 
20 

3o 
4o 
5o 

8.21  189 
8.21  319 
8.21  447 
8.21  576 
8.21  7o3 
8.21  83i 

8.21  195 
8.21  324 
8.21  453 

8.21  58i 
8.21  709 
8.21  837 

o    4 

5o 
4o 
3o 
20 

IO 

49  o 

10 
20 
3o 
4o 
5o 

8.i5  39i 
8.15538 
8.15685 
8.i5832 
8.i5978 

8.l6  123 

8.i5  395 
8.15543 
8.i5  690 

8.i5836 
8.i5  982 
8.16  128 

o   11 

5o 
4o 
3o 

20 
IO 

57  o 

IO 
20 

3o 
4o 
5o 

8.21  958 
8.22  085 

8.22  211 
8.22  337 

8.22463 
8.22  588 

8.21  964 

8.22  091 
8.22  217 
8.22  343 
8.22  469 
8.22  595 

o    3 

5o 
4o 
3o 

20 
IO 

50  o 

10 
20 

3o 
4o 
5o 

8.16  268 
8.i64i3 
8.i6557 

8.16  700 
8.i6843 
8.16986 

8.16  273 
8.16417 
8.i656i 
8.16  705 
8.i6848 
8.  16  991 

o   10 

5o 
4o 
3o 

20 
10 

58  o 

IO 
20 

3o 
4o 
5o 

8.22  713 
8.22838 

8.22  962 

8.  23  086 

8.23  210 

8.23  333 

8.22  720 
8.22844 
8.22968 

8.23  092 
8.232i6 
8.23  339 

o    2 

5o 
4o 
3o 
20 

IO 

51  o 

10 
20 

3o 
4o 
5o 

8.17  128 
8.17  270 
8.  17  4i  i 
8.17  552 
8.  17  692 
8.17882 

8.17  i33 
8.17275 
8.17416 
8.17557 
8.17697 
8.i7837 

o     9 

5o 
4o 
3o 
20 

IO 

59  o 

10 

20 
3o 

4o 
5o 

8.23456 
8.23578 
8.23  700 
8.23  822 
8.23944 
8.24065 

8.23462 
8.23  585 
8.23  707 
8.23  829 
8.23  950 
8.24  071 

o     1 

5o 
4o 
3o 
20 

10 

52  o 

10 

20 

3o 

8.17971 
8.18  no 
8.18  249 

8.  T*  3s7 

8.17976 
8.18  ii5 
8.18254 
8.18  392 

o     8 

5o 
4o 
3o    7 

60  o 

8.24  186 

8.24  192 

o     0 

L.  Cos.     L.  Cotg1. 

"     ' 

L.  Cos. 

L.  Cotg". 

"    ' 

126 


89°. 


FUNCTIONS    OF    SMALL    ANGLES. 
1°. 


/     II 

L.  Sin. 

L.Tang. 

r      ti 

L.  Sin. 

L.  Tang. 

0    o 

10 
20 

8.24  186 
8.243o6 
8.24426 

8.24  192 
8.243i3 
8.24433 

o  60 
5o 

4o 

7  Jo 

4o 
5o 

8.29  3oo 
8.29  407 
8.29  5i4 

8.29  309 
8.29416 
8.29  523 

3o 

20 
IO 

3o 
4o 
5o 

8.24546 
8.24665 

8.24785 

8.24553 

8.24  672 
8.24  791 

3o 

20 

10 

8    o 

IO 
20 

8.29  621 
8.29727 
8.  29  833 

8.29  629 
8.29  736 
8.29842 

o  52 

5o 
4o 

1     o 

10 
20 

8.24903 
8.25  022 
8.25  i4o 

8.24  910 
8.25  029 
8.25  i47 

o  59 

5o 
4o 

3o 
4o 
5o 

8.29939 
8-.3oo44 
8.3o  150 

8.29  947 
8.3oo53 
8.3o  i58 

3o 

20 
10 

3o 

4o 
5o 

8.25258 
8.25  375 
8.25493 

8.25265 
8.25  382 
8.25  500 

3o 
20 
10 

9    o 

IO 
20 

8.30255 
8.3o359 
8.3o464 

8.3o263 
8.3o368 
8.3o473 

o  51 

5o 

4o 

2    o 

10 

20 

8.25  609 
8.25  726 
8.25842 

8.25  616 
8.25  733 
8.25849 

o  58 

5o 
4o 

3o 

4o 
5o 

8.3o568 
8.30672 
8.30776 

8.3o577 
8.3o68i 
8.30785 

3o 

20 
IO 

3o 
4o 
5o 

8.25958 
8.26074 
8.26  189 

8.25965 
8.26081 
8.26  196 

3o 

20 
10 

10  o 

IO 

20 

8.30879 
8.3o983 
8.3i  086 

8.3o888 
8.3o  992 
8.3i  095 

o  50 

5o 

4o 

3     o 

10 
20 

8.263o4 
8.26419 
8.26533 

8.26  3i2 
8.26426 
8.2654i 

0    57 

5o 

4o 

3o 
4o 
5o 

8.3i  188 
8.3i  291 
8.3i  393 

8.  3  1  198 
8.  3  1  3oo 

8.  3  1  4o3 

3o 

20 
10 

3o 

4o 
5o 

8.26648 
8.26  761 
8.26875 

8.26655 
8.26  769 
8.26882 

3o 

20 
10 

11  o 

10 
20 

8.3i  4g5 
8.3i  597 
8.3i  699 

8.3i  505 
8.3i  606 
8.3i  708 

o  49 

5o 

4o 

4    o 

10 
20 

8.26988 
8.27  101 
8.27  214 

8.26  996 
8.27  109 

8.27  221 

o  56 

5o 

4o 

3o 
4o 
5o 

8.3i  800 
8.3i  901 

8.32  002 

8.  3  1  809 
8.3i  911 

8.32  012 

3o 
20 

IO 

3o 
4o 
5o 

8.27  326 

8.  27  438 
8.27  550 

8.27334 
8.27446 
8.27558 

3o 

20 
IO 

12  o 

10 

20 

8.32  io3 
8.32  2o3 
8.32  3o3 

8.32  112 

8.32213 
8.32  3i3 

o  48 

5o 

4o 

5     o 

10 

20 

8.27  661 
8.27773 
8.27883 

8.27  669 
8.27  780 
8.27  891 

o  55 

5o 
4o 

3o 
4o 
5o 

8.324o3 
8.325o3 
8.32  602 

8.324i3 
8.32  5i3 
8.32612 

3o 

20 
1C 

3o 
4o 
5o 

8.27  994 
8.28  104 
8.28  215 

8.28002 
8.28  112 

8.28  223 

3o 

20 
IO 

13  o 

10 

20 

8.32  702 
8.32  801 
8.32  899 

8.32  711 
8.32811 
8.32  909 

o  47 

5o 
4o 

6    o 

10 
20 

8.28324 
8.28434 
8.28  543 

8.28  332 

8.28442 

8.2855i 

o  54 

5o 
4o 

3o 
4o 
5o 

8.32  998 
8.  33  096 
8.33  195 

8.33  008 
8.33  106 
8.33205 

3o 

20 
IO 

3o 
4o 
5o 

8.28652 
8.28  761 
8.28  869 

8.28660 
8.28  769 
8.28  877 

3o 
20 

10 

14  o 

10 

20 

8.33292 
8.  3339o 

8.33488 

8.33  3o2 
8.334oo 
8.33498 

o  46 

5o 
4o 

7    o 

10 

20 

8.28977 
8.29085 
8.29  193 

8.28  986 
8.29  094 

8.29  2OI 

o  53 

5o 
4o 

3o 
4o 
5o 

8.33  585 
8.33682 
8.33779 

8.33  595 
8.  33  692 
8.33789 

3o 
20 

10 

3o 

8  .  29  3oo 

8.29  3og 

3o52 

15  o 

33875 

8.33886 

o  45 

L.  Cos. 

L.  cotg. 

"    ' 

L.  Cos. 

L.  Cotg. 

FUNCTIONS    OP    SMALL    ANGLES 
1°. 


/      tr 

L.  Sin. 

L.  Tang. 

,       „ 

L.  Sin. 

L.  Tang. 

15  o 

10 
20 

8.33876 
8.33972 
8.34068 

8.33  886 
8.33982 
8.34078 

o  45 

5o 

4o 

22  3o 
4o 
5o 

8.38  oi4 
8.38  101 
8.38  189 

8.38  026 
8.38  n4 
8.38  202 

3o 

20 
10 

3o 
4o 
5o 

8.34  i64 
8.34260 
8.34355 

8.34174 
8.34270 
8.34366 

3o 
20 

10 

23  o 

10 

20 

8.38  276 
8.38363 
8.3845o 

8.38289 
8.38376 
8.38463 

o  37 

5o 
4o 

16  o 

10 

eo 

8.3445o 
8.34546 
8.3464o 

8.3446i 
8.34556 
8.3465i 

o  44 

5o 
4o 

3o 

4o 
5o 

8.38  537 
8.38  624 
8.38  710 

8.38  550 
8.38636 
8.38  723 

3o 

20 
10 

3o 
4o 
5o 

8.34735 
8.3483o 
8.34924 

8.34746 
8.3484o 
8.34935 

3o 
20 
10 

24  o 

10 

20 

8.38  796 
8.38882 
8.38,968 

8.38  809 
8.38895 
8.38981 

o  36 

5o 
4o 

17  o 

I  O 

20 

8.35oi8 
8.35  112 
8.35206 

8.35  029 
8.35  123 
8.35217 

o  43 

5o 
4o 

3o 

4o 
5o 

8.39  o54 
8.39  1  39 

8.39  225 

8.39067 
8.39  i53 
8.39238 

3o 
20 

IO 

3o 
4o 
5o 

8.35  299 
8.  35392 
8.35485 

8.353io 
8.354o3 
8.35497 

3o 
20 
10 

25  o 

10 

20 

8.39  3io 
8.39395 
8.39480 

8.39323 
8.39408 
8.39493 

o  35 

5o 
4o 

18  o 

10 

20 

8.35578 
8.  35  671 
8.35764 

8.35  590 
8.35682 
8.35775 

o  42 

5o 

4o 

3o 
4o 
5o 

8.39565 
8.39  649 
8.39734 

8.39578 
8.39663 
8.39747 

3o 
20 

IO 

3o 
4o 
5o 

8.35856 
8.35948 
8.36  o4o 

8.35867 
8.35959 
8.36o5i 

3o 
20 
10 

26  o 

IO 

20 

8.39818 
8.3g  902 
8.39986 

8.39832 
8.39  916 
8.4o  ooo 

o  34 

5o 
4o 

19  o 

10 
20 

8.36  i3i 
8.36223 
8.363i4 

8.36  i43 
8.3623s 
8.36326 

o  41 

5o 
4o 

3o 
4o 
5o 

8.4o  070 
8.4o  i53 
8.40237 

8.4oo83 
8.4o  i67 
8.4o25i 

3o 
20 

IO 

3o 
4o 
5o 

8.364o5 
8.36496 
8.36587 

8.36417 
8.365o8 
8.36  599 

3o 

20 
IO 

27  o 

10 

20 

8.4o  320 
8.4o4o3 
8.4o486 

8.4o334 
8.4o4i7 
8.4o  500 

o  33 

5o 
4o 

20  o 

10 

20 

8.36678 
8.36768 
8.36858 

8.36  689 
8.36  780 
8.36870 

o  40 

5o 
4o 

3o 
4o 
5o 

8.40669 
8.4o65i 
8.40734 

8.4o583 
8.4o665 
8.4o  748 

3o 
20 

IO 

3o 
4o 
5o 

8.36948 
8.37038 
8.37  128 

8.36  960 
8.37050 
8.37  i4o 

3o 
20 

IO 

28  o 

IO 
20 

8.40816 
8.40898 
8.40980 

8.4o83o 
8.40913 
8.4o  995 

o  32 

5o 
4o 

21  o 

10 

20 

8.37217 
8.37  3o6 
8.37395 

8.37  229 
8.373i8 
8.374o8 

o  39 

5o 
4o 

3o 

4o 
5o 

8.4i  062 
8.4i  1  44 
8.4i  225 

8.4i  o77 
8.4i  i58 
8.4i  240 

3o 

20 
10 

3o 
4o 
5o 

8.37484 
8.37  573 
8.37662 

8.  37497 
8.37585 
8.37674 

3o 
20 

10 

29  o 

IO 

20 

8.4i  307 
8.4i  388 
8.4i  469 

8.4i  32i 
8.4i4o3 

8.4i484 

o  31 

5o 
4o 

22  o 

10 

20 

8.37750 
8.37838 
8.37  926 

8.37762 
8.3785o 
8.37938 

o  38 

5o 
4o 

3o 
4o 
5o 

8.4i  55o 
8.4i  63i 
8.4i  711 

8.4i  565 
8.4i  646 
8.4i  726 

3o 

20 
IO 

3o 

8.38oi4 

8.38026 

3o37 

30  o 

Mi  792 

Mi  8o7 

o'30 

L.  Cos. 

L.  Cotg. 

n       , 

L.  Cos. 

L.  Cotg. 

'     " 

128 


88°. 


FUNCTIONS    OF    SMALL    ANGLES. 

1°. 


/    " 

L.  Sin.    L.Tang. 

/    ft 

L.  Sin.    L.Tang. 

30  o 

10 

20 
3o 
4o 
5o 

8.4i  79- 
8.4i  872 
8.4i  962 
8.42  o32 

8.42  112 
8.42  192 

8.4i  807 
8.4i  887 
8.4i  967 
8.42  o48 
8.42  127 
8.42  207 

o  30 

5o 
4o 
3o 
20 
10 

37  3o 
4o 
5o 

8.45  267 
8.4534i 
8.454i5 

8.45  285 
8.45  359 
8.45433 

3o 
20 

IO 

38  o 

IO 

20 
3o 
4o 
5o 

8.45489 
8.45563 
8.45637 
8.  45  710 
8.  45  784 
8.45857 

8.  455o7 
8.4558i 
8.45655 

8.45728 
8.45  802 
8.45875 

o  22 

5o 
4o 
3o 

20 
IO 

31  o 

10 
20 
3o 
4o 
5o 

8.42  272 

8.4235i 
8.4243o 

8.42  5io 
8.42689 
8.42667 

8.42  287 
8.42  366 
8.42446 
8.42625 
8.42  6o4 
8.42683 

o  29 

5o 
4o 
3o 
20 

10 

39  o 

IO 
20 

3o 
4o 
5o 

8.45  930 
8.46oo3 
8.46076 
8.46  149 
8.46  222 
8.46  294 

8.45948 
8.46  021 
8.46094 

8.46  167 
8.46  240 
8.46  3i2 

oSl 

5o 
4o 
3o 

20 
10 

32  o 

10 
20 

3o 
4o 
5o 

8.42  746 
8.42825 
8.42903 
8.42982 
8.43  060 
8.43  i38 

8.42  762 
8.42  84o 
8.42  919 

8.42  997 
8.43075 
8.43x54 

o  28 

5o 
4o 
3o 
20 
10 

40  o 

10 

20 
3o 
4o 
5o 

8.46  366 
8.46439 
8.465n 

8.46583 
8.4665s 
8.46727 

8.46  385 
8.46457 
8.46529 

8.46602 
8.46674 
8.  46  745 

o  20 

5o 
4o 
3o 

20 
10 

33  o 

10 
20 
3o 
4o 
5o 

8.43  2ltJ 

8.43293 
8.  4337i 

8.43448 
8.43526 
8.436o3 

8.43232 
8.43  3o9 
8.  43387 

8.43464 
8.43542 
8.436i9 

o  27 

5o 
4o 
3o 
20 

10 

41  o 

IO 

20 
3o 
4o 
5o 

8.46  799 
8.46870 
8.46942 
8.47oi3 

8.47084 
8.47  i55 

8.46817 
8.46889 
8.46960 

8.47o3a 
8.47  io3 
8.47  i?4 

o  19 

5o 
4o 
3o 

20 
10 

34  o 

10 

20 
3o 
4o 
5o 

8.4^680 
8.43757 
8.43834 
8.43910 
8.43987 
8.44o63 

8.43  696 
8.43773 
8.4385o 

8.43  927 
8.44oo3 
8.  44  080 

o  26 

5o 
4o 
3o 

20 
10 

42  o 

IO 

20 
3o 
4o 
5o 

8.47226 
8.47297 
8.47368 

8.47439 
8.47  509 
8.47  58o 

8.47245 
8.473i6 
8.47387 
8.47458 
8.47528 
8.47  599 

o  18 

5o 
4o 
3o 
20 

10 

35  o 

10 

20 

3o 
4o 
5o 

8.44  139 
8.44216 
8.44292 
8.44367 
8.44443 
8.445i9 

8.44  1  56 
8.44232 
8.443o8 

8.44384 
8.4446o 
8.44536 

o  25 

5o 
4o 
3o 

20 
IO 

43  o 

IO 

20 
3o 

4o 

5o 

8.47  650 
8.47  720 
8.47  790 
8.47  860 
8.47  930 
8.48  ooo 

8.47  669 
8.47  74o 
8.47810 

8.47  880 

8.4795° 
8.48  020 

o  17 

5o 
4o 
3o 
20 

10 

36  o 

10 
20 
3o 
4o 
5o 

8.44594 
8.44669 

8.44745 
8.44820 
8.44895 
8.44969 

8.446n 
8.44686 
8.44762 
8.44837 
8.44912 
8.44987 

o  24 

5o 
4o 
3o 

20 
IO 

44  o 

IO 
20 

3o 
4o 
5o 

8.48  069 
8.48  i39 
8.48  208 

8.48  278 
8.48  347 
8.484i6 

8.48090 
8.48  i59 
8.48  228 

8.48  298 
8.48  367 
8.48436 

o  16 

5o 
4o 
3o 
20 

10 

37  o 

10 

20 

3o 

8.45o44 
8.45  119 
8.45  193 

8.45  -j-;- 

8.45  061 
8.45  i36 
8.45  210 
8.45  285 

o  23 

5o 

4o 

3o22 

45  o 

J.  48  485 

M8  5o5 

o   15 

L.  Cos.    L.  Cotg. 

"    ' 

L.  Cos. 

L.  Cotg. 

"     ' 

88°. 


129 


FUNCTIONS   OF   SMALL   ANGLES 
1°. 


,  „ 

L.  Sin. 

L.  Tang. 

,     „ 

L.Sin. 

L.  Tang. 

45  o 

10 

20 

8.48  485 
8.48  554 
8.48622 

8.48  5o5 
8.48574 
8.48643 

o  15 

5o 
4o 

52  3d 

4o 
5o 

8.5i  48o 
8.5i  544 
8.  5  1  609 

8.5i  5o3 
8.5i  568 
8.5i  632 

3o 

20 
IO 

3o 
4o 
5o 

8.48  691 
8.48  760 
8.48828 

8.48  711 
8.48  780 
8.48849 

3o 

20 
IO 

53  o 

10 

20 

8.5i  673 
8.5i  737 
8.5i  801 

8.5i  696 
8.5i  760 

8;5l  824 

o    7 

5o 
4o 

46  o 

10 
20 

8..  48  896 
8.48  965 
8.49033 

8.48917 
8.48985 
8.49.053 

o  14 

5o 

4o 

3o 
4o 
5o 

8.5;  864 
8.5i  928 
8.5i  992 

8.5i  888 
8.5i  952 
8.52  oi5 

3o 
20 

IO 

3o 

4o 
5o 

8.49  101 
8.49  169 
8.49236 

8.49  121 
8.49  189 
8.49257 

3o 

20 
10 

54  o 

IO 

20 

8.52o55 
8.52  119 
8.52  182 

8  .  52  079 
8.52  i43 
8.52  206 

o    6 

5o 
4o 

47  o 

10 

20 

8.49  3o4 
8.49372 
8.49439 

8.49325 

8.49  393 
8.49460 

o  13 

5o 

4o 

3o 
4o 
5o 

8.52245 
8.523o8 
8.5237i 

8.52  269 
8.52  332 
8.52  396 

3o 
20 

10 

3o 
4o 
5o 

8.49  5o6 
8.49574 
8.49641 

8.49528 
8.49  595 
8.49  662 

3o 

20 
10 

55  o 

IO 

20 

8.52434 
8.52  497 
8.52  56o 

8.52  459 

8.52  522 

8.52  584 

o    5 

5o 

4o 

48  o 

10 
20 

8.49  708 

8.49775 
8.49842 

8.49  729 
8.49  796 
8.49863 

o  12 

5o 
4o 

3o 
4o 
5o 

8.52623 
8.52685 
8.52748 

8.52647 
8.52  710 

8.52  772 

3o 

20 
10 

3o 
4o 
5o 

8.49908 
8.49975 
8.5oo42 

8.49930 

8.49997 
8.5oo63 

3o 

20 
10 

56  o 

IO 

20 

8.52  810 
8.52872 
8.52935 

8.52  835 
8.52897 
8.52  960 

o    4 

5o 
4o 

49  o 

10 

20 

8.5o  108 
8.5o  i74 
8.5o24i 

8.5o  i3o 
8.5o  196 
8.50263 

o   11 

5o 
4o 

3o 
4o 
5o 

8.52  997 
8.53o59 
8.53  121 

8.53022 
8.53o84 
8.53  i46 

3o 
20 

10 

3o 
4o 
5o 

8.5o  307 
8.5o373 
8.5o439 

8.5o329 
8.50395 
8.5o46i 

3o 
20 

IO 

57  o 

IO 
20 

8.53  i83 
8.53245 
8.533o6 

8.53  208 
8.53  270 
8.53  332 

o    3 

5o 

4o 

50  o 

10 
20 

8.5o  5o4 
8.5o  570 
8.5o636 

8.5o527 
8.5o593 
8.5o658 

o   10 

5o 

4o 

3o 
4o 
5o 

8.53368 
8.53429 
8.53491 

8.53  393 
8.53455 
8.535i6 

3o 

20 
IO 

3o 
4o 
5o 

8.5o  701 
8.5o  767 
8.5o832 

8.5o  724 
8.5o  789 
8.5o855 

3o 
20 

10 

58  o 

IO 
20 

8.53552 
8.536i4 
8.  53  675 

8.53578 
8.53639 
8.53  700 

o    2 

5o 
4o 

51  o 

10 
20 

8.50897 
8.5o963 
8.5i  028 

8.5o  920 
8.5o985 
8.5i  o5o 

o     9 

5o 

4o 

3o 
4o 
5o 

8.53736 
8.53797 
8.53858 

8.53762 
8.53823 
8.53884 

3o 

20 
10 

3o 
4o 
5o 

8.5i  092 
8.5i  i57 

8.5l  222 

8.5i  u5 
8.5i  180 
8.5i  245 

3o 

20 
10 

59  o 

10 
20 

8.53919 
8.  53  979 
8  .  54  o4o 

8.53945 
8.54oo5 
8.  54  066 

o     1 

5o 

4o 

52  o 

10 

20 

8.5i  287 
8.5i  35i 
8.5i  4i6 

8.5i  3io 
8.5i  374 
8.5i  439 

o     8 

5o 

4o 

3o 
4o 
5o 

8.54  ioi 
8.54  161 
8.54  222 

8.54  127 
8.54  187 
8.  54  s48 

3o 
20 

10 

3o 

8.5i  48o 

.5i  5o3 

3o   7 

60  o 

8.542* 

8.54  3o8 

>    0 

L.  Cos. 

L.  Cotg. 

,,    * 

L.Cos. 

L.  Cotg. 

' 

130 


88°. 


TABLE    IV 

FOUR-PLACE 
NAPERIAN    LOGARITHMS 


NAPERIAN    LOGARITHMS. 

LOGARITHMS    OF    POWERS    OF    10. 


Num. 

Log. 

Num. 

Log. 

10 

2.3O26 

.  i 

3~.6974 

IOO 

4.6o52 

.01 

5.3948 

1000 

6.9078 

.001 

7.0922 

IOOOO 

9.2103 

.0001 

^.7897 

IOOOOO 

1  1  .5129 

.00001 

72.4871 

IOOOOOO 

i3.8i55 

.00000  I 

a.  1845 

IOOOOOOO 

16.1181 

.000000  I 

17.8819 

IOOOOOOOO 

18.4207 

.00000001 

19.5793 

I  OOOOOOOOO 

20.7233 

.00000000  1 

21  .2767 

Num. 

Log. 

Num. 

Log. 

LOGARITHMS  OF  NUMBERS  FROM  i  TO  10. 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

1.0 

o.oooo 

OIOO 

0198 

0296 

0392 

o488 

o583 

o677 

0770 

0862 

.1 

.2 

.3 

0.0953 

0.1823 

0.2624 

io44 
1906 
2700 

n33 

1989 
2776 

1222 
2070 
2852 

i3io 

2l5l 

2927 

1398 

223l 

3ooi 

i484 

23ll 

3o75 

i57o 
2390 
3i48 

i655 
2469 

3221 

i74o 
2546 
3293 

.4 
.5 
.6 

0.3365 
o.4o55 
0.4700 

3436 

4l2I 

4762 

35o7 

4i87 
4824 

3577 
4253 
4886 

3646 
43i8 
4947 

37i6 

4383 
5oo8 

3784 
4447 
5o68 

3853 
45n 

5i28 

392O 

4574 
5i88 

3988 
4637 
5247 

•  7 

.8 

•9 

o.53o6 

o.5878 
0.6419 

5365 
5933 
6471 

5423 
5988 
6523 

548i 
6o43 
6575 

5539 
6098 
6627 

5596 
6i52 
6678 

5653 
6206 
6-729 

57io 
6259 
678o 

5766 
63i3 
683i 

5822 

6366 
6881 

2.0 

0.6931 

6981 

7o3i 

7080 

7129 

7178 

•722-7 

7275 

7324 

7372 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

NAPERIAN    LOGARITHMS. 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

2.0 

2.1 

2.2 
2.3 

0.6931 

6981 

7o3i 

7080 

7129 

7178 

7227 

7275 

7324 

7372 

0.7419 
o.7885 
0.8329 

7467 

793° 
8372 

75i4 
7975 
84i6 

756i 
8020 
8459 

7608 
8o65 
85o2 

7655 
8109 

8544 

77OI 

8i54 
8587 

7747 
8198 
8629 

7793 
8242 
8671 

7839 
8286 
87i3 

2.4 
2.5 
2.6 

o.8755 
0.9163 
o.9555 

8796 
9203 
9^94 

8838 
9243 
9632 

8879 

9282 

967o 

8920 
9822 
9708 

8961 
936i 
9746 

9002 
9400 
9783 

9042 
9439 

982I 

9o83 
9478 
9858 

9I23 

95i7 
9895 

2.7 

2.8 

2.9 

3.0 

3.i 

3.2 

3.3 

0.9933 
1.0296 
1.0647 

9969 
o332 
0682 

6006 
0367 
0716 

6o43 
o4o3 
0750 

6080 
o438 
0784 

61  16 
o473 
0818 

Ol52 

o5o8 
o852 

018,8 
o543 
0886 

0225 

o578 
o9i9 

6260 
o6i3 
o953 

1.0986 

1019 

io53 

1086 

1119 

n5i 

n84 

1217 

I249 

1282 

I.i3i4 
i.i632 
i.i939 

1  346 
i663 
1969 

i378 
1694 

2000 

i4io 
1725 
2o3o 

1  442 
i756 
2060 

i474 
i787 
2090 

i5o6 
181-7 

21  19 

i537 

1  848 

2l49 

i569 

1878 

2179 

1600 
I9o9 
2208 

3.4 
3.5 
3.6 

1.2238 
1.2528 

1.2809 

2267 
2556 
2837 

2296 

2585 
2865 

2326 

26i3 

2892 

2355 
2641 
2920 

2384 
2669 
2947 

24i3 
2698 
29-75 

2442 
2726 

3002 

2470 
2754 

3o29 

2499 

2782 
3o56 

3.7 
3.8 
3.9 

4.0 

4.i 

4.2 

4.3 

i.3o83 
i.335o 
i.36io 

3uo 
3376 
3635 

3i37 
34o3 
366i 

3i64 
3429 
3686 

3i9i 

3455 
3712 

3218 
348  1 
3737 

3244 
35o7 
3762 

3271 

3533 

3788 

3297 
3558 
38i3 

3324 

3584 
3838 

1.3863 

3888 

3913 

3938 

3962 

3987 

4012 

4o36 

4o6i 

4o85 

i.4no 
i.435i 
1.4586 

4i34 
4375 
4609 

4i59 
4398 
4633 

4i83 

4422 

4656 

4207 
4446 
4679 

423i 
4469 
4702 

4255 
4493 

472D 

4279 
45i6 

4748 

43o3 

454o 
477<> 

4327 
4563 
4793 

4.4 
4.5 
4.6 

i.48i6 
i.5o4i 
1.5261 

4839 
5o63 
5282 

486i 
5o85 
53o4 

4884 
5107 
5326 

4907 
5129 

5347 

4929 
5i5i 
5369 

49^1 

5i73 
5390 

4974 
5i95 
54i2 

4996 
5217 
5433 

5oi9 
5239 
5454 

4-7 
4.8 

4-9 
5.0 

5.i 

5.2 

5.3 

1.5476 
1.5686 
1.5892 

5497 
57o7 
SgiS 

55i8 
5728 
5933 

5539 

5748 
5953 

556o 
5769 
5974 

558i 
579o 
5994 

56o2 
58io 
6oi4 

5623 
583i 
6o34 

5644 
585i 
6o54 

5665 
5872 
6o74 

1.609^ 

6n4 

6i34 

6i54 

6i74 

6194 

6214 

6233 

6253 

6273 

1.6292 
i.6487 
1.6677 

63i2 
65o6 
6696 

6332 
6525 
67i5 

635i 
6544 
6734 

637i 
6563 
6752 

63go 
6582 
6771 

6409 
6601 
6-790 

6429 
6620 

6808 

6448 
6639 
6827 

6467 
6658 
6845 

5.4 
5.5 
5.6 

1.6864 
1.7047 
1.7228 

6882 
7066 
7246 

6901 
7o84 
7263 

6919 
7102 
7281 

6938 
7120 
7299 

6956 
7i38 
73i7 

6974 
7i56 
7334 

6993 
7i74 
7352 

7011 

7I92 

737o 

•7029 

721O 

7387 

5-7 
5.8 
5.9 

6.0 

i.74o5 
1.7579 
i.775o 

7422 

7596 
7766 

744o 
76i3 

7783 

7457 
763o 
7800 

7475 
7647 
7817 

74g2 
7664 
7834 

75o9 
768i 
785i 

7527 

7699 

7867 

7544 
77i6 

7884 

756i 
7733 
79oi 

1.7918 

7934 

7951 

7967 

7984 

8001 

8017 

8o34 

8o5o 

8066 

0 

1 

2    3 

4 

5 

6 

7 

8 

9 

i33 


NAPERIAN    LOGARITHMS. 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

6.0 

1.7918 

7934 

795i 

7967 

79»4 

8001 

8017 

8o34 

8o5o 

8066 

6.1 

6.2 

6.3 

i.8o83 
.8245 
.84o5 

8099 
8262 
8421 

8116 

8278 
8437 

8i32 
8294 
8453 

8i48 
83io 
8469 

8i65 
8326 

8485 

8181 
8342 
85oo 

8i97 
8358 
85i6 

82i3 
8374 
8532 

8229 

839o 

8547 

6.4 
6.5 
6.6 

.8563 

.8718 
.8871 

8579 
8733 
8886 

8594 
8749 
89oi 

8610 
8764 
89i6 

8625 
8779 
8931 

864i 
8795 
8946 

8656 
8810 
896i 

8672 

8825 
8976 

8687 
884o 
899i 

87o3 
8856 
9oo6 

6.7 
6.8 
6.9 

.9021 
.9169 
.93i5 

9o36 
9184 
933o 

9o5i 
9i99 
9344 

9o66 

92l3 

9359 

9081 
9228 
9373 

9o95 
9242 
9387 

9i  10 

9257 
9402 

9I25 
9272 

94i6 

9i4o 
9286 
943o 

9i55 
93oi 
9445 

7.0 

.9459 

9473 

9488 

95o2 

95i6 

953o 

9544 

9559 

9573 

9587 

7-i 
7.2 
7.3 

.9601 
.974i 
.9879 

96i5 
9755 
9892 

9629 

9769 
99o6 

9643 
978a 

9920 

9657 
9796 
9933 

967i 
98io 

9947 

9685 
9824 
996i 

9699 

9838 
9974 

97i3 
985i 
9988 

9727 
9865 

OOOI 

7-4 
7.5 

7-6 

2.  001  5 

2.0149 
2.0281 

0028 
0162 
0295 

0042 
0176 
o3o8 

oo55 

oi89 

0321 

0069 

O2O2 

o334 

0082 
02  1  5 
o347 

oo96 

O229 

o36o 

oio9 
0242 
o373 

OI22 
0255 

o386 

oi36 
0268 
o399 

7-7 
7.8 

7-9 

2.0412 

2.o54i 

2.0669 

o425 
o554 
0681 

o438 
0567 
o694 

o45i 
o58o 
0707 

o464 
0592 
0719 

o477 
o6o5 
0732 

o49o 

0618 
o744 

o5o3 
o63i 

o757 

o5i6 
o643 
o769 

o528 
o656 

0-782 

8.0 

2.0794 

0807 

o8i9 

o832 

o844 

o857 

o869 

0882 

o894 

o9o6 

8.1 

8.2 

8.3 

2  .0919 
2.I04I 

2.u63 

ogSi 
io54 
1175 

o943 
1066 

1187 

o956 
1078 
1199 

0968 
1090 

I2II 

o98o 
1  1  02 

1223 

0992 

1  1  14 
1235 

ioo5 
1126 

I247 

IOI7 

n38 
1258 

IO29 

i  i5o 

I27O 

8.4 
8.5 
8.6 

2.1282 

2.l4oi 

2.i5i8 

1294 
1412 
1529 

i3o6 

1424 
i54i 

i3i8 
i436 
i552 

i33o 

1  448 
1  564 

1  342 

i459 
1576 

i353 
i47i 
i587 

i365 

i483 
i599 

i377 
i494 
1610 

i389 
i5o6 
1622 

8.7 
8.8 
8.9 

2.i633 

2.1748 
2.1861 

1  645 
1759 

1872 

i656 
1770 

i883 

1668 
1782 
1894 

1679 
i793 
I9o5 

i69i 
1804 
I9i7 

1702 
i8i5 

I928 

I7i3 
182-7 
i939 

I725 

i838 
i95o 

i736 
1849 
1961 

9.0 

2.  1972 

i983 

i994 

2006 

2017 

2028 

2o39 

2o5o 

2061 

2072 

9.1 
9.2 
9.3 

2.2083 
2.2192 
2.23OO 

2094 

22O3 
23ll 

2IO5 
22l4 
2322 

21  16 

2225 
2332 

2127 

2235 

2343 

2i38 

2246 

2354 

2148 

2257 

2364 

2l59 

2268 

2375 

2I7O 
2279 

2386 

2181 
2289 
2396 

9.4 
9.5 
9.6 

2.24O7 
2.25l3 

2.2618 

2418 

2523 

2628 

2428 

2534 
2638 

2439 

2544 
2649 

245o 
2555 
2659 

2460 
2565 
2670 

247r 
2576 
2680 

2481 
2586 

269O 

2492 

2597 
2701 

2502 
2607 
2711 

9-7 
9.8 
9.9 

2.2721 
2.2824 
2.2925 

2732 
2834 
2935 

2742 
2844 
2946 

2752 
2854 
2g56 

2762 
2865 
2966 

2773 

2875 

2976 

2783 

2885 
2986 

2?93 
2895 
2996 

28o3 
29o5 
3oo6 

28l4 
29l5 

3oi6 

10.0 

2.3026 

3i26 

3224 

3322 

34i8 

35i4 

36o9 

37o3 

3796 

3888 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

1 34 


TABLE    V 

FOUR-PLACE    LOGARITHMS 
OF    NUMBERS 


FOUR-PLACE    LOGARITHMS. 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

"IT 

oooo 

o43 

086 

128 

170 

212 

•B^^^^B 

253 

•BBBMH 

294 

334 

~ 

1  1 

4i4 

453 

492 

53i 

569 

607 

645 

682 

719 

755 

12 

792 

828 

864 

899 

934 

969 

100^ 

ro38 

1072 

1  106 

i3 

ii39 

I73 

206 

239 

271 

3o3 

335 

367 

399 

43o 

i4 

46i 

492 

523 

553 

584 

6i4 

644 

673 

7o3 

732 

i5 

1761 

79° 

818 

847 

875 

903 

93i 

959 

987 

2OI, 

16 

20 

4i 

068 

o95 

122 

1  48 

i75 

2OI 

227 

253 

279 

ll 

3o4 

33o 

355 

38o 

4o5 

43o 

455 

48o 

5o4 

529 

18 

553 

577 

60  1 

625 

648 

672 

695 

718 

742 

765 

'9 

788 

810 

833 

856 

878 

900 

923 

945 

967 

989 

20 

3oio 

032 

o54 

o75 

o96 

118 

i39 

1  60 

181 

201 

21 

222 

243 

263 

284 

3o4 

324 

345 

365 

385 

4o4 

22 

424 

444 

464 

483 

5o2 

522 

54  1 

56o 

579 

598 

23 

617 

636 

655 

674 

692 

7' 

I 

729 

747 

766 

784 

24 

802 

820 

838 

856 

874 

892 

9°9 

927 

945 

962 

25 

3979 

997 

4or4 

4o3i 

4o48 

4o65 

4082 

4o99 

4n6 

4i33 

26 

4i5o 

1  66 

i83 

200 

216 

232 

249 

265 

281 

298 

27 

3i4 

33o 

346 

362 

378 

393 

4o9 

425 

44o 

456 

28 

472 

487 

502 

5i8 

533 

548 

564 

579 

594 

6o9 

29 

624 

639 

654 

669 

683 

698 

7i3 

728 

742 

757 

30 

477i 

786 

800 

8i4 

829 

843 

857 

87, 

886 

9oo 

3i 

9i4 

928 

942 

955 

969 

983 

997 

5oi  i 

5o24 

5o38 

32 

5o5i 

o65 

o79 

092 

io5 

n9 

i3a 

i45 

i59 

I72 

33 

1  85 

198 

21  I 

224 

237 

25o 

263 

276 

289 

302 

34 

3i5 

328 

34o 

353 

366 

378 

39i 

4o3 

4i6 

428 

35 

544  1 

453 

465 

478 

490 

502 

5i4 

527 

539 

55i 

36 

563 

575 

587 

599 

611 

623 

635 

647 

658 

67o 

37 

682 

694 

7o5 

717 

729 

74o 

752 

763 

775 

786 

38 

798 

8o9 

821 

832 

843 

855 

866 

877 

888 

899 

39 

911 

922 

933 

944 

955 

966 

977 

988 

999 

6010 

40 

••••••i 

6021 

•MM^^^M 

o3i 

•••••• 

042 

••HM^M 

o53 

••••Mi 

o64 

o75 

o85 

o96 

IO7 

117 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

PP 

38 

32 

28 

35 

22 

21 

19 

18 

17 

16 

.1 

.2 

7-6 

1! 

>    5-6 

5-o 

.2 

4-4 

4-2 

3-8 

.2 

3-6 

3-4 

3-2 

•3 

11.4 

9.e 

8.4 

7-5 

•3 

6.6 

6-3 

5-7 

•3 

5-4 

4.8 

•4 

»5-2 

I2.i 

II.2 

IO.O 

•4 

8.8 

8.4 

7.6 

•4 

7.2 

6.8 

6.4 

•  5 

19.0 

j6.c 

>    14.0 

12.5 

•5 

I.O 

10.5 

9-5 

•5 

9.0 

8-5 

8.0 

6 

22.8 

19.2 

16.8 

15.0 

.6 

3-2 

12.6 

11.4 

.6 

10.8 

10.2 

9.6 

•7 

26.6 

22.4 

19.6 

J7-5 

•  7 

5-4 

'4-7 

'3-3 

•  7 

12.6 

ii.  9 

II.  2 

.8 

30-4 

25.6 

22.4 

20.0 

.8 

7.6 

1  6.'  8 

'5-2 

.8 

14.4 

13-6 

12.8 

•9    34-2 

28.8   25.2   22.5 

•9 

9.8   18.9 

J5-  3   '4-4 

1 36 


FOUR-PLACE    LOGARITHMS. 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

40 

6021 

o3i 

042 

o53 

064 

M^M 

07 

HIM 

5 

o85 

••—  ^— 
096 

I07 

117 

4i 

128 

1  38 

i49 

160 

I70 

180 

191 

201 

212 

222 

42 

232 

243 

253 

263 

274 

284 

294 

3o4 

3i4 

325 

43 

335 

345 

355 

365 

375 

385 

395 

4o5 

4i5 

425 

44 

435 

444 

454 

464 

474 

484 

493 

5o3 

5x3 

522 

45 

6532 

542 

55i 

56i 

57i 

58o 

59o 

599 

6o9 

618 

46 

628 

637 

646 

656 

665 

675 

684 

693 

702 

712 

47 

721 

73o 

739 

749 

758 

767 

776 

785 

794 

8o3 

48 

812 

821 

83o 

839 

848 

857 

866 

875 

884 

893 

49 

902 

911 

92O 

928 

937 

946 

955 

964 

972 

981 

50 

6990 

998 

7oo7 

7016 

7024 

7o33 

7042 

7o5o 

7°59 

7o67 

5i 

7076 

o84 

o93 

IOI 

I  IO 

118 

126 

i35 

i43 

152 

52 

1  60 

168 

I77 

i85 

i93 

202 

210 

218 

226 

235 

53 

243 

25l 

269 

267 

275 

284 

292 

3oo 

3o8 

3i6 

54 

324 

332 

34o 

348 

356 

364 

372 

38o 

388 

396 

55 

74o4 

4l2 

4i9 

427 

435 

443 

45i 

459 

466 

474 

56 

482 

490 

497 

5o5 

5i3 

52O 

528 

536 

543 

55i 

57 

559 

566 

574 

582 

589 

597 

6o4 

612 

6i9 

627 

58 

634 

642 

649 

657 

664 

672 

679 

686 

694 

701 

59 

709 

716 

723 

73i 

738 

745 

752 

760 

767 

774 

60 

7782 

789 

796 

8o3 

810 

818 

825 

832 

839 

846 

61 

853 

860 

868 

875 

882 

88 

9 

896 

9o3 

9io 

917 

62 

924 

93i 

938 

945 

952 

959 

966 

973 

98o 

987 

63 

993 

8000 

8oo7 

8oi4 

802  1 

8028 

8o35 

8o4i 

8o48 

8o55 

64 

8062 

o69 

o75 

082 

089 

o96 

102 

io9 

116 

122 

65 

8l29 

i36 

149 

i56 

162 

i69 

176 

182 

189 

66 

i95 

202 

209 

2l5 

222 

228 

235 

2 

4r 

248 

254 

67 

261 

267 

274 

280 

287 

293 

299 

3o6 

3l2 

319 

68 

325 

33i 

338 

344 

35i 

357 

363 

37o 

376 

382 

69 

388 

395 

4oi 

4o7 

4i4 

420 

426 

432 

439 

445 

70 

45i 

457 

463 

4?o 

476 

482 

488 

••^•••v 

494 

•i^^M  1 

5oo 

••^^•V 

5o6 

N 

0 

2 

3 

4 

5 

6 

7 

8 

9 

PP 

15 

14    13 

12 

IO 

9 

8 

7 

6 

., 

1.5 

1.4    1.3 

1.2 

.! 

i.i 

I.O 

0.9 

.1 

0.8 

0.7 

0.6 

.2 

3-o 

2.8     2.6 

2.4 

.2 

2.2 

2.0 

1.8 

.2 

1.6 

1.4 

1.2 

•3 

4-5 

4-2    3-9 

3-6 

•  3 

3-3 

3-° 

2.7 

•3 

2.4 

2,1 

1.8 

•4 

6.0 

5-6    5-2 

4-8 

•  4 

4-4 

4.0 

3-6 

•4 

3-2 

2.8 

2.4 

•5 

7-5 

7.0    6.5 

6.0 

•5 

5-o 

4-5 

•  5 

4.0 

3-5 

3-o 

.6 

9.0 

8.4    7-8 

7.2 

.6 

6.6 

6.0 

5-4 

.6 

4.8 

4.2 

3-6 

•  7 

10.5 

9.8    9.1 

8.4 

•7 

7-7 

7.0 

6-3 

•7 

5-6 

4-9 

4.2 

.8 

I2.O 

1  1.  2    10.4 

9.6 

.8 

8.8 

8.0 

7.2 

8 

6.4 

5-6 

4.8 

•9    i3-5 

126   11.7   10.8    .9    9.9 

i37 


FOUR-PLACE    LOGARITHMS. 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

70 

845i 

~ 

463 

470 

476 

482 

488 

494 

5oo 

5o6 

71 

5i3 

5i9 

525 

53i 

537 

543 

549 

555 

56i 

567 

72 

573 

579 

585 

5oi 

597 

6o3 

6o9 

6i5 

621 

627 

73 

633 

639 

645 

65i 

657 

663 

669 

675 

681 

686 

74 

692 

698 

704 

710 

716 

722 

727 

733 

739 

745 

75 

875! 

756 

762 

768 

774 

779 

785 

79i 

797 

802 

76 

808 

8i4 

820 

825 

83i 

837 

842 

848 

854 

859 

77 

865 

871 

876 

882 

887 

893 

899 

9o4 

9io 

9i5 

78 

92I 

927 

932 

938 

943 

949 

954 

96o 

965 

971 

79 

976 

982 

987 

993 

998 

9oo4 

9oo9 

9oi5 

9O2O 

9025 

80 

9o3i 

o36 

042 

047 

o53 

o58 

o63 

o69 

o74 

079 

81 

o85 

o9o 

o96 

IOI 

106 

112 

117 

122 

128 

i33 

82 

1  38 

i43 

i49 

1  54 

i59 

i65 

170 

i75 

180 

186 

83 

|9I 

i96 

2OI 

206 

212 

2I7 

222 

227 

232 

238 

84 

243 

248 

253 

258 

263 

269 

274 

279 

284 

289 

85 

9294 

299 

3o4 

3o9 

3i5 

320 

325 

33o 

335 

34o 

86 

345 

35o 

355 

36o 

365 

37o 

375 

38o 

385 

390 

87 

39! 

j 

4oo 

4o5 

4io 

4i5 

420 

425 

43o 

435 

44o 

88 

445 

45o 

455 

46o 

465 

469 

474 

479 

484 

489 

89 

494 

499 

5o4 

5o9 

5i3 

5i8 

523 

528 

533 

538 

90 

9542 

547 

552 

557 

562 

566 

57i 

576 

58i 

586 

91 

59( 

) 

595 

600 

6o5 

6o9 

6i4 

6i9 

624 

628 

633 

92 

63( 

J 

643 

647 

652 

657 

661 

666 

671 

675 

680 

93 

685 

689 

694 

699 

7o3 

7o8 

7i3 

717 

722 

727 

94 

73i 

736 

74  1 

745 

75o 

754 

759 

763 

768 

773 

95 

9777 

782 

786 

79i 

795 

800 

8o5 

8o9 

8i4 

818 

96 

823 

827 

832 

836 

84  1 

845 

85o 

854 

859 

863 

97 

868 

872 

877 

881 

886 

89o 

894 

899 

-9o3 

9o8 

98 

9I2 

9i7 

081 

926 

9 

3o 

934 

939 

943 

948 

952 

99 

956 

96i 

965 

969 

974 

978 

983 

987 

99i 

996 

100 

oooo 

oo4 

oo9 

oi3 

OI7 

022 

026 

o3o 

o35 

o4o 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

PP 

7 

6 

5 

4 

.1 

0.7 

0.6 

.1          0.5 

0.4 

.2 

M 

12 

.2              1.0 

0.8 

•3 

2.1 

i  8 

•3          *-5 

1.2 

.4 

2.8 

2.4 

.4             2.0 

1.6 

•  5 

3-5 

3  o 

•  3          2.5 

2.O 

.6 

4.2 

3-6 

.6          3.0 

2.4 

•  7 

4-9 

4.2 

•7          3-5 

2.8 

.8 

5-6 

4.8 

.8          4-0 

3-2 

-9 

5-4 

•9           4-5          3-6 

i38 


TABLE   VI 

FOUR-PLACE    LOGARITHMS 

OF  THE 

TRIGONOMETRIC    FUNCTIONS 

TO   EVERY   TEN    MINUTES 


POUR-PLACE    LOGARITHMIC   FUNCTIONS. 


O       ' 

L.  Sin. 

d. 

L.Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

0    o 

10 
20 

3o 
4o 
5o 

7 

7 

8 
8 

.4637 
.7648 

.9408 
.o658 
.1627 

son 
1760 
1250 

969 
792 
669 
580 

5" 

458 
4i3 

578 

3011 
1761 
i249 
969 
792 
670 
580 
5" 

457 
4i5 
378 
348 
322 
300 
281 
263 
249 
235 
223 

213 

202 
194 
l85 
I78 
I7I 
l65 
158 
»54 
I48 

2.5363 

2.2352 

2.0591 
1.9342 
i.8373 

o.oooo 

0  .  OOOO 

o.oooo 

o.oooo 
o.oooo 
o.oooo 

0 

o 
o 

0 

0 

I 

0 
0 

o 

0 

I 

0 

I 

0 

o 

I 

0 

I 
I 
I 

o    90 

5o 

4o 

3o 
20 

10 

7.4637 
7.7648 

7-9409 

8.o658 
8.1627 

1     o 

10 

20 

3o 
4o 
5o 

8 
8 
8 

8 
8 
8 

.2419 
.3o88 
.3668 

.4179 
.4637 
.5o5o 

8.2419 
8.3089 
8.3669 

8.4i8i 

8.4638 
8.5o53 

i.758i 
i  .691  i 
i.  633  i 

1.5819 
1.5362 

i  .4947 

9-9999 
0.9999 

9.9999 

9.9999 
9.9998 
9.9998 

o    89 

5o 
4o 

3o 

20 
IO 

2    o 

10 
20 

3o 
4o 
5o 

8 
8 
8 

8 
8 
8 

.5428 
.5776 
.6097 

.6397 
.6677 
.6940 

348 
321 
300 
280 
263 
248 
235 

222 
212 
2O2 
I92 

8.543i 
8.5779 
8.6101 

8.64oi 
8.6682 
8.6945 

.4569 
.4221 
.3899 

.3599 
.33i8 
.3o55 

9-9997 
9-9997 
9.9996 

9.9996 
9.9995 
9-999^ 

o    88 

5o 
4o 

3o 
20 

IO 

3    o 

10 

20 

3o 
4o 
5o 

8 
8 
8 

8 
8 
8 

.7188 
.7423 
.7645 

.7867 
.8069 
.8261 

8.7194 
8.7429 
8.7652 

8.7865 
8.8067 
8.8261 

.2806 
.2571 
.2348 

.2i35 
.i933 
.1739 

9-9994 
9.9993 
9.9993 

9.9992 
9.9991 
9-999° 

o    87 

5o 
4o 

3o 
20 

IO 

4    o 

IO 
20 

3o 
4o 
5o 

8 
8 
8 

8 
8 
8 

.8436 
.86i3 

.8783 

.8946 
.9104 
.9266 

185 
I77 
170 
l63 
158 
IS2 
M7 

8.8446 
8.8624 
8.8795 

8.8960 
8.9118 
8.92-72 

.i554 
.1376 

.1205 

.  io4o 

.0882 
.0728 

9.9989 
9.9989 
9.9988 

9.9987 
9.9986 
9.9985 

0 

I 

o    86 

5o 
4o 

3o 

20 
10 

5 

0 

8.9403 

8.9420 

i.o58o 

9.9983 

o    85 

L 

.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L.Sin. 

d. 

'         O 

PP 

.2 

•3 
•4 

:I 
1 

348 

300 

263 

2 

3 
4 

I 
I 

235 

213 

185 

.i 

.2 

•3 
•4 

J 

:! 

171 

158      147 

S3 
104.4 

139.2 
174.0 
208.8 

»j 

g 

9o 

120 
150 
1  80 

210 
240 

26.3 

52-6 
78.9 

105.2 
i3i-5 
157-8 

184., 
210.4 

• 

23-5 
47.0 
70.5 

94.0 
117.5 
141.0 

164.5 
i88.c 

21.3 

42.  c 

63-9 

85.2 
106.5 

127.1 

149.1 
170.4 

18.5 

37-o 
55-5 

74.0 
92-5 

III.O 

129.5 
148.0 
166.5 

17.1 
34-2 
5i-3 

68.4 
85-5 

102.6 

119.7 
136.8 

15.8     14.7 

31.6      29.4 

47-4        44-i 

63.2        58.8 
79-o        73-5 
94.8       88.2 

no.  6      102.9 
126.4      117-6 

142.2      i32-3 

POUR-PLACE   LOGARITHMIC   FUNCTIONS. 


O        ' 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

5    o 

IO 

20 

3o 

4o 
5o 

8.9403 
8.9545 
8.9682 

8.9816 
8.9945 
9.0070 

142 

137 
'34 
129 
125 

122 

"5 
"3 
109 
107 
104 

102 

99 
97 
95 
93 

89 
87 
85 
84 
82 
80 

79 
78 
76 
75 
73 
73 

8.9420 
8.9563 
8.9701 

8.9836 
8.9966 
9.0093 

143 
138 
135 
130 
127 
123 

120 

"4 
in 

1  08 
105 
104 

101 

98 

97 
94 
93 

89 
87 
86 

84 
82 
81 
80 
78 
77 
76 
74 

i.o58o 
1.0437 
1.0299 

i.oi64 
i.oo34 
0.9907 

9.9983. 
9.9982 
9.998i 

9.998o 
9-9979 
9-9977 

2 

I 
I 
2 
I 
I 
2 
I 
2 
2 
I 
2 
2 

o    85 

5o 

3o 
20 

IO 

6    o 

IO 

20 

3o 
4o 
5o 

9.0192 
9.081  i 
9.0426 

9.0539 
9.0648 
9.0755 

9.0216 
9.o336 
9.o453 

9.0667 
9.0678 
9.0786 

0.9784 
0.9664 
0.9547 

0.9433 
0.9322 
0.9214 

9-9976 
9-9975 
9-9973 
9.9972 
9.9971 
9.9969 

o    84 

5o 
4o 

3o 

20 
10 

7    o 

IO 

20 

3o 
4o 
5o 

9.0859 
9.0961 
9.  1060 

9.n57 

9-1252 

9.i345 

9.0891 
9.o995 
9.  1096 

9.1194 
9.1291 
9.  i  385 

0.9109 
0.9005 
0.8904 

0.8806 
0.8709 
o.86i5 

9.9968 
9.9966 
9.9964 

9.9963 
9.9961 
9.9959 

o    83 

5o 

4o 

3o 
20 

IO 

8    o 

IO 

20 

3o 

4o 
5o 

9.i436 

9.  1612 

9.1697 
9.1781 
9.i863 

9.1478 
9.1569 
9.i658 

9.1745 
9.i83i 
9.i9i5 

0.8522 

o.843i 
0.8342 

0.8255 
0.8169 
o.8o85 

9.9958 
9.9956 
9.9954 

9.9952 
9.995o 
9.9948 

I 
2 
2 
2 
2 
2 
2 
2 
2 
2 
2 
2 
2 

o    82 

5o 
4o 

3o 

20 
10 

9    o 

IO 

20 

3o 
4o 
5o 

9.1943 
9.2022 
9.2100 

9.2176 
9.2324 

9-1997 
9.2O78 
9.2i58 

9.2236 
9.23i3 

9.2889 

o.8oo3 
0.7922 
0.7842 

0.7764 
0.7687 
0.7611 

9.9946 
9.9944 
9.9942 

9.9940 
9.9938 
9  .  9936 

o   81 

5o 

4o 

20 
IO 

10    o 

9.2397 

9.2463 

o.7537 

9.9934 

o    80 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L.  Sin. 

d. 

'       o 

PP     138 

"5 

"7 

.1 

.2 

•3 
•4 

•9 

104  97 

89 

.1 

.2 

•3 
•4 

I 

84 

78          73 

'       I3'J 
27.  o 

•3        4'-4 

•4        55-2 
•  5       69.0 
.6       82.8 

.7       96.6 
.8      110.4 
.9      124.2 

12.5 
25.0 
37-5 

50.0 
62.5 
75-° 

87-5 

1OO.O 

112.5 

11.7 
23-4 

46.8 
58.5 
70.2 

8..g 
93.6 

105-3 

10.4  9.7 
20.8  19.4 
31.2  29.1 

41.6  38.8 
52-0  48.5 
62.4  58.2 

72.8  67.9 
83.2  77.6 

93.6  87.3 

8.9 
17.8 
26.7 

35-6 
44-5 
53-4 

62.3 
71.2 

80.  i 

25-2 

33-6 
42.0 
50.4 

58.8 
67.2 

7-8          7-3 
15.0        14.6 
23.4        21.9 

31.2        29.2 
39-o        36.5 
46.8        43.8 

54-6        51.1 
62.4        58.4 
70.  2        65.  7 

i4r 


FOUR-PLACE   LOGARITHMIC    FUNCTIONS. 


O       ' 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L.  Cos. 

d. 

10    o 

10 

20 

3o 
4o 
5o 

9 
9 

9 

9 
9 
9 

.2397 
.2468 
.2538 

.2606 
.2674 
.2740 

71 
7o 

68 

68 
66 

9.  2463 
9.2536 
9  .2609, 

9.2680 
9.2750 
9.2819 

73 
73 
7i 
70 
69 
68 
66 
67 
65 
64 
63 
63 
61 
61 
61 
59 
59 
58 
57 
57 
56 
55 
55 
54 
53 
53 
53 
5» 
52 
5i 

0.7537 
0.7464 
o.739i 

0.7320 
0.7250 
0.7181 

9.9934 
9.993i 
9.9929 

9.9927 
9.9924 
9.9922 

3 

2 
2 

3 

2 

o    80 

5o 
4o 

3o 

20 
10 

11     o 

IO 
20 

3o 
4o 
5o 

9 

9 
9 

9 
9 
9 

.2806 
.2870 
.2934 

.2997 
.3o58 
.3119 

64 
64 
63 
61 
61 

9.2887 
9.2953 
9.3o2o 

9.3o85 
9.3i49 

9.32I2 

0.7113 
0.7047 
0.6980 

o.69i5 

o.685i 

0.6788 

9.9919 
9.9917 
9.9914 

9.9912 
9.9909 
9.9907 

3 

2 

3 

2 

3 

2 

o    79 

5o 
4o 

3o 
20 

IO 

12    o 

10 
20 

3o 

4o 
5o 

9 
9 
9 

9 
9 
9 

•3i79 
.3238 
.3296 

.3353 
.34io 
.3466 

59 
58 
57 
57 
56 

9.3275 
9.3336 
9.3397 

9.3458 
9.35  i  7 
9.3576 

0.6725 
0.6664 
o.66o3 

O.6542 
0.6483 
0.6424 

9.9904 
9.9901 
9.9899 

9.9896 
9.9893 
9.9890 

3 
3 

2 

3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
4 
3 
3 
4 

o    78 

5o 

4o 

3o 
20 

10 

13    o 

10 

20 

3o 
4o 
5o 

9.352i 
9.3575 
9.3629 

9.3682 
9.3734 
9.3786 

55 
54 
54 
53 
52 
52 
5i 
5° 
50 
49 
49 
48 

47 

9.3634 
9.369i 
9.3748 

9.38o4 
9.3859 
9.39i4 

0.6366 
o.63o9 
0.6252 

o.6i96 
o.6i4i 
0.6086 

9.9887 
9.9884 
9.9881 

9.9878 
9.9875 
9.9872 

o    77 

5o 
4o 

3o 

20 
10 

14    o 

10 

20 

3o 
4o 
5o 

9.3837 
9.3887 
9.3937 

9.3986 
9.4o35 
9.4o83 

9.3968 
9.4021 
9.4o74 

9.4127 
9.4178 
9.4a3o 

o.6o32 
o.5979 
o.5926 

0.5873 

0.5822 

0.5770 

9.9869 
9.9866 
9.9863 

9.9859 
9.9856 
9.9853 

o    76 

5o 

4o 

3o 
20 

IO 

15 

0 

9.4i3o 

9.4281 

o.57i9 

9.9849 

o    75 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L.  Sin. 

d. 

'        o 

PP 

.2 

•3 
•4 

:l 

7i 

68 

66 

.1 

.2 

•3 

•4 

:! 
:i 

•9 

64          61            58 

.1 

.2 

•3 
•4 

:! 

:i 

55 

53           5i 

7-1 
14.2 
21.3 

28.4 

33 

Si 

6.8 
13-6 
20.4 

27.2 
34-o 
40.8 

47.6 
54-4 
61.2 

6.6 

!|! 

26.4 
33-o 
39-6 

46.2 
52.8 
59-4 

6.4        6.1         5.8 

12.8          12.2           II.  6 
19.2          l8.3           17.4 

25.6         24.4          23.2 
32.0         30.5          29.0 
38.4         36.6           34.8 

44.8         42.7          40.6 
51.2         48.8          46.4 

57-  6       54-9        52-2 

5-5 

II.  0 

16.5 

22.0 
27-5 

33-o 

38.5 

44.0 

49-5 

5-3          5-i 

10.6            10.2 

iS-9        '5-3 

21.2           20.4 
26.5          25.5 
31.8          30.6 

37-  *        35^7 
42.4       40.8 

47-7        45-9 

142 


FOUR-PLACE    LOGARITHMIC    FUNCTIONS. 


O         ' 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L. 

Cos. 

d. 

15    o 

10 

20 

3o 

4o 
5o 

9.4i3o 

9.4177 
9.4223 

9.4269' 
9.43i4 
9.4359 

47 
46 
46 
45 
45 
44 
44 
44 

42 

43 
42 
4i 
41 
4i 
40 
40 
40 
39 
39 
38 
38 
37 
38 
36 
37 
36 
36 
35 
36 
35 

9.4281 
9.433i 

9.438i 

9.443o 

9.4479 
9.4527 

50 
50 

49 
49 
48 
48 
47 
47 
47 
46 
46 
45 
45 
'  45 
44 
44 
44 
43 
43 
42 
42 
42 
42 
4i 
4* 
40 
40 
40 
40 
40 

o.57i9 
0.5669 
0.5619 

0.5570 
0.5521 

0.5473 

9.9849 
9.9846 
9.9843 

9.9839 
9.9836 
9.9832 

3 
3 
4 
3 
4 

o    75 

5o 
4o 

3o 
20 

IO 

16    o 

10 
20 

3o 
4o 
5o 

9.44o3 
9-4447 
9.4491 

9.4533 
9.4576 
9.4618 

9.4575 
9.4622 
9.4669 

9.4716 
9.4762 

9.4808 

o.5425 
0.5378 
o.533i 

0.5284 
0.5238 
0.5192 

9.9828 
9.9825 
9.9821 

9.9817 
9.9814 
9.9810 

3 
4 
4 
3 
4 

o    74 

5o 

4o 

3o 

20 
IO 

17    o 

IO 

20 

3o 
4o 
5o 

9-4659 
9.4700 
9.4741 

9.4781 
9.4821 
9.4861 

9.4853 
9.4898 
9.4943 

9.4987 
9-5o3i 
9.5075 

o.5i47 

O.  5  I  02 

o.5o57 

o.5oi3 
0.4969 
0.4925 

9.9806 
9.9802 
9.9798 

9.9794 
9.9790 
9.9786 

4 
4 
4 
4 
4 
4 

o    73 

5o 

4o 

3o 
20 

IO 

18    o 

10 
20 

3o 

4o 
5o 

9.4900 
9.4939 
9.4977 
9.  5oi5 
9-5o52 
9.5090 

9.5u8 
9«5i6i 
9.52o3 

9.5245 
9.5287 
9.5329 

0.4882 
0.4839 
0.4797 
o.4755 
o.47i3 
0.4671 

9.9782 
9.9778 
9-9774 

9-977° 
9.9765 

9-976* 

4 
4 
4 
4 
5 
4 

o    72 

5o 
4o 

3o 

20 
IO 

19    o 

IO 

20 

3o 
4o 
5o 

9.5126 
9.5i63 
9.5199 

9.5235 
9.5270 
9.53o6 

9.5370 
9.54n 
9.545i 

9.5491 
9.553i 

9.5571  , 

o.463o 
0.4589 
0.4549 

0.4509 
0.4469 
0.4429 

9.9757 
9.9752 
9-9748 

9.9743 
9.9739 
9.9734 

5 
4 
5 
4 
5 

o    71 

5o 

4o 

3o 
20 

10 

20 

0 

9.534i 

9.56n 

0.4389 

9.973o 

o    70 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L.  Sin. 

d. 

'         O 

PP 

2 

3 

4 
5 
.6 

•  7 
.8 

•9 

49         47 

45 

.1 

.2 
«3 

•4 

•  5 
.6 

44 

43 

41 

.1 

.2 

•  3 

•4 
5 
6 

! 

40 

38 

36 

4-9        4-7 
9.8        9.4 
14.7       14.1 

19.6       18.8 
24-5       23.5 
29.4       28.2 

34-3       32.9 
39-2       37-6 
44-  1       42.3 

4-5 
9.0 
13-5 

18.0 
22.5 
27.0 

3'-  5 
30.0 

4°-5 

ti 

13-2 
17.6 

22.  0 
26.4 

30.8 
35-2 

tl 

12.9 

17.2 
21.5 
25.8 

30.1 

34-4 

38.7 

t: 

12.3 

i6.4 
20.5 
24.6 

28.7 

32.8 

4.0 

8.0 

12.0 

16.0 

20.0 
24.0 

28.0 
32.0 
36.0 

3-8 

7-6 
11.4 

15-2 
19.0 

22.8 

26.6 
30-4 

34-2 

3-6 

7-2 

10.8 

14.4 
18.0 

21.6 

25.2 
28.8 
32-4 

i43 


POUR-PLACE    LOGARITHMIC    FUNCTIONS. 


O         ' 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.  Cotg. 

L. 

Cos. 

d. 

20    o 

10 

20 

3o 
4o 
5o 

9.534i 
9.5375 
9.5409 

9.5443 
9.5477 
9.55io 

34 
34 
34 
34 
33 
33 
33 
33 
32 
32 
3» 
32 
3i 
3» 
3<> 
3* 
3° 
3° 
29 
3° 
29 
29 

29 
28 
28 
28 
28 
28 
27 
27 

9«56i  i 
9.565o 
9.  5689 

9.5727 
9.5766 
9.58o4 

39 
39 
38 
39 
38 
38 
37 
38 
37 
37 
37 
36 
36 
36 
36 
36 
35 
36 
35 
34 
35 
34 
35 
34 
34 
33 
34 
33 

34 

• 
33 

o.4389 
o.435o 
o.43n 

0.4273 
0.4234 
o.4i96 

9.973o 
9.9725 
9.9721 

9.9716 
9.9711 
9.9706 

5 
4 
5 
5 
5 

o    70 

5o 
4o 

3o 

20 
10 

21    o 

10 
20 

3o 

4o 
5o 

9.5543 
9.5576 
9.5609 

9.564i 
9.5673 
9.5704 

9-5842 
9.5879 
9.5917 

9.5954 
9.599i 
9.6028 

o.4i58 

0.4  I  21 

o.4o83 

o.4o46 
o.4oo9 
o.3972 

9.9702 

9.9697 
9.9692 

9.9687 

9.9682 

9.9677 

5 

5 
5 
5 
5 

o    69 

5o 
4o 

3o 

20 
IO 

22    o 

10 
20 

3o 
4o 
5o 

9.5736 
9.5767 
9.5798 

9.5828 
9.5859 
9.5889 

9.6064 
9.6100 
9.6i36 

9.6172 
9.6208 
9.6243 

o.3936 
o.39oo 
0.3864 

0.3828 
o.3792 
o.3757 

9.9672 
9.9667 

9.9661 
9.9656 

9.9651 
9.9646 

5 
6 

5 
5 
5 

o    68 

5o 
4o 

3o 
20 

IO 

23    o 

10 

20 

3o 
4o 
5o 

9.5919 
9.5948 
9.5978 

9.6007 
9.6o36 
9.6o65 

9.6279 
9.63i4 
9.6348 

9.  6383 
9.6417 
9.  6452 

0.3721 
0.3686 
0.3652 

0.3617 
0.3583 
0.3548 

9.9640 

9.9635 
9.9629 

9.9624 

9.9618 

9.9613 

5 
6 
5 
6 
5 

o    67 

5o 
4o 

3o 

20 
10 

24    o 

IO 

20 

3o 

4o 
5o 

9.6093 
9.6121 
9.6149 

9.6177 
9.6205 
9.6232 

9.6486 
9.6520 
9.  6553 

9.6587 
9.662o 
9.6654 

o.35i4 
o.348o 
0.3447 

o.34i3 
o.338o 
0.3346 

9.9607 

9.9602 

9.9596 

9.9590 
9.9584 
9.9579 

5 
6 
6 
6 
5 

o    66 

5o 
4o 

3o 

20 
10 

25 

0 

9.6259 

9.6687 

o.33i3 

9.9573 

o    65 

L.  Cos. 

d. 

L.Cotg. 

d. 

L.  Tang. 

L.  Sin. 

d. 

'        0 

PP 

.  i 

.2 

•3 
•4 

•  7 
.8 

•9 

39 

37 

35 

,i 

.2 

•3 

•4 
•5 
.6 

1 

•9 

34 

33 

32 

2 

3 
4 

1 

31 

3° 

29 

1:1 

11.7 

15-6 
'9-5 
23-4 

27-3 
3'  ? 
35-i 

3-7 
7-4 
n.  i 

14.8 
18.5 

22.2 

25-9 
29.6 

3-5 
7.0 
10.5 

14.0 
17-5 

21.0 

24-5 
28.0 

3'-5 

U 

10.2 

I3.6 
17.0 
20.4 

23.8 
27.2 

30.6 

U 

9-9 

III 

19.8 

23.1 
26.4 

20.  7 

E 

9.6 

12.8 

16.0 
19.2 

22.4 

25.6 

n 

9-3 
12.4 

;i:i 

£J 

27.9 

3-o 
6.0 
9.0 

12.0 
15-0 

18.0 

21.0 
24.0 

I1 

8.7 

ii.  6 
i4-5 
17-4 

20.3 
23.2 
26.1 

1 44 


FOUR-PLACE    LOGARITHMIC    FUNCTIONS. 


0           ' 

L.  Sin. 

d. 

L.Tang. 

d. 

L.  Cotg. 

L. 

Cos. 

d. 

25    o 

10 

20 

3o 

4o 
5o 

9 
9 
9 

9 
9 
9 

.6269 

.6286 
.63i3 

.634o 
.6366 
.6392 

27 
27 
27 

26 
26 
26 
26 
26 
25 
26 

25 

9.6687 
9.6720 
9.6762 

9.6786 
9.6817 
9.6860 

33 
32 
33 
32 
33 
32 
32 
32 
31 
32 

32 
31 

3» 

30 

30 
3* 
30 

30 
30 
29 
30 
29 
30 
29 
29 

o.33i3 
0.3280 
0.3248 

o.32i5 
o.3i83 
o.3i5o 

9.9673 
9.9667 
9.9661 

9.9666 
9.9549 
9.9643 

6 
6 
6 

6 
6 

o    65 

5o 
4o 

3o 
20 
10 

26    o 

10 
20 

3o 
4o 
5o 

9 
9 
9 

9 
9 
9 

.64i8 
.6444 
.6470 

.6496 
.6621 
.6546 

9.6882 
9.6914 
9.6946 

9.6977 
9.7009 
9.7040 

o.3n8 
o.3o86 
o.3o54 

o.3o23 
0.2991 
0.2960 

9.9537 
9.953o 
9.9524 

9.9618 
9.9612 
9.9606 

7 
6 
6 
6 
7 

o    64 

5o 
4o 

3o 

20 
10 

27    o 

10 
20 

3o 
4o 
5o 

9 
9 
9 

9 
9 
9 

.6670 
.6696 
.6620 

.6644 
.6668 
.6692 

24 
25 
25 
24 
24 
24 

9.7072 
9.7103 
9  .  7  i  34 

9.7166 
9.7i96 

9.7226 

0.2928 
0.2897 
0.2866 

0.2835 
0.2804 

0.2774 

9.9499 
9.9492 
9.9486 

9.9479 
9.9473 
9.9466 

7 
6 

7 
6 

7 

o    63 

5o 

4° 
3o 
20 
10 

28    o 

10 
20 

3o 
4o 
5o 

9 
9 
9 

9 
9 
9 

.6716 
.6740 
.6763 

.6787 
.6810 
.6833 

24 
23 
24 
23 
23 

9-7257 
9.7287 
9.7317 

9.7348 
9.7408 

o.2743 
0.2713 
0.2683 

0.2662 
0.2622 
0.2692 

9.9459 
9.9453 
9.9446 

9.9439 
9.9432 
9.9426 

6 
7 
7 
7 
7 

o   62 

5o 
4o 

3o 

20 
10 

29    o 

10 

20 

3o 
4o 
5o 

9 
9 
9 

9 
9 
9 

.6856 
.6878 
.6901 

.6923 
.6946 
.6968 

22 

23 
22 

23 
22 

9.7438 
9.7467 
9.7497 

9.7626 
9.7666 
9.7585 

0.2662 
0.2533 
o.25o3 

0.2444 
0.24  i  5 

9.9418 
9.9411 
9  .  94o4 

9.9397 
9.9390 
9.9383 

7 
7 
7 

7 
7 

o    61 

5o 
4o 

3o 
20 
10 

30 

0 

9 

.6990 

9.7614 

0.2386 

9.9375 

o    60 

L 

.  Cos. 

d. 

L.  Cotg. 

d. 

L.Tang. 

L.  Sin. 

d. 

'         O 

PP 

.2 

•3 
•4 

:J 

•9 

28 

27 

26 

.1 

.2 

•3 
•4 

:i 

25 

24       23 

.1 

22 

7 

6 

2.8 

5-6 
8.4 

II.  2 
14.0 

16.8 

19.6 
22.4 

2-7 

ti 

10.8 
13-5 
16.2 

18.9 

21.6 

24-3 

2.6 

10.4 

13.0 
15.6 

18.2 

20.8 

23-4 

2-5 

7-5 

10.0 

12.5 
15-0 

17-5 

20.0 

22-5 

2.4      2.3 

4.8      4.6 
7.2      6.9 

9.6     9.2 

12.0          II.5 
14.4         13.8        : 

16.8       16.1      i 
19.2       18.4 
21.6       20.7 

2.2 

n 

8.8 

II.O 

13-2 

19.8 

0.7 
1.4 

2.1 

2.8 

3-5 
4.2 

4.9 
5-6 
6-3 

0.6 

1.2 

1.8 

2-4 

5-4 

i45 


FOUR-PLACE    LOGARITHMIC    FUNCTIONS 


o 

r 

L.  Sin. 

d. 

L.  Tang. 

d. 

L.Cotg. 

L.  Cos. 

d. 

30    o 

10 
20 

3o 
4o 
5o 

9.6990 

9.7OI2 
9.7088 

9.7O55 
9.7076 
9.7097 

22 
21 
22 
21 
21 
21 
21 
21 
21 
20 
21 
20 
20 
20 
20 
20 
20 
19 
19 
20 

19 

18 

18 

18 
18 

9-76i4 
9.7644 
9-7673 

9-77°i 
9-773o 
9.7759 

30 
29 
28 

29 
29 

0.2886 
0.2356 
0.2827 

0.2299 
0.2270 

O.224l 

9.9375 
9.9868 
9.9861 

9.9353 
9.9346 
9-9338 

7 
7 

8 

7 
8 

o    60 

5o 
4o 

3o 

20 
10 

31    o 

10 
20 

3o 
4o 
5o 

9.7II8 

9.7160 

9.7181 
9.7201 
9.  7222 

9.7788 
9.7816 
9.7845 

9.7878 
9.7902 
9.7980 

29 
28 

29 
28 
29 
28 
28 
28 
28 
28 
28 
27 
28 
28 
27 
28 

27 
28 
27 
27 
27 
27 
27 
27 

O.22I2 
0.2184 

o.2i55 

0.2127 
0.2098 
0.2070 

9.933i 
9.9323 
9.9815 

9.9808 
9.9800 
9.9292 

7 
8 
8 
7 
8 
8 

o    59 

5o 
4o 

3o 

20 

10 

32    o 

10 

20 

3o 

4o 

5o 

9.7242 
9.7262 
9.7282 

9.7802 
9.7822 
9.7342 

9.7958 
9.7986 
9.8014 

9.8042 
9.8070 
9.8097 

O.2O42 
O.2OI4 
0.1986 

0.1958 

o.  1980 
o.  1908 

9.9284 

9.9268 

9.9260 
9.9252 
9.9244 

8 
8 
8 
8 
8 

o    58 

5o 
4o 

3o 
20 

10 

33    o 

10 

20 

3o 
4o 

5o 

9.736i 
9.738o 
9  .  74oo 

9.7419 
9.  7438 
9.7457 

9.8125 
9.8i53 
9.8180 

9.8208 
9.8235 
9.8268 

0.1875 

0.1847 
o.  1820 

0.1792 
o.  1765 

o.i737 

9.9286 
9.9228 
9.9219 

9.9211 
9.9208 
9.9194 

8 

9 
8 
8 
9 

o    57 

5o 

4o 

3o 
20 
10 

34   o 

10 
20 

3o 

4o 
5o 

9.7476 
9.7494 
9.75i3 

9.753i 
9.755o 
9.7568 

9.8290 
9.83i7 
9.  8344 

9.837i 
9.8898 
9.8425 

0.  I7IO 

0.1688 

o.i656 

o.  1629 
o.  1602 
o.  i575 

9.9186 
9.91-77 
9.9169 

9.9160 
9.9i5i 
9.9142 

9 
8 

9 
9 
9 
8 

o    56 

5o 
4o 

3o 

20 
IO 

35 

0 

9.7586 

9.8452 

27 

o.i548 

9.9184 

o    55 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L.Sin. 

d. 

'         O 

PP 

2 

3 

4 

j 

29           28 

27 

.2 

•3 

•4 
•  5 
.6 

•9 

22 

21 

20 

.2 

•3 
•4 

19 

8            7 

2.9        2.8 
5-8         5-6 
8.7         8.4 

II.  6         II.  2 

14-5      14-0 
17.4      16.8 

20.3      19.6 
23.2      22.4 
26.1       25.2 

2.7 

10.8 
16.2 
18.9 

21.6 

24-3 

2.2 

a 

8.8 

II.  0 

13-2 

\7'.6 

IQ.8 

2.1 

8-4 
10.5 

12.6 

i6!8 
18.9 

2.0 
4-0 

6.0 

8.0 

10.0 
12.  0 

14.0 

16.0 

18.0 

'•9 

3-8 
5-7 

7.6 
9-5 
11.4 

13-3 
15-2 

0.8          0.7 
1.6          1.4 
2.4          2.1 

3-2               2.8 

4-°          3-5 
4.8          4.2 

5-6          4-9 
6.4          5.6 

i46 


POUR  PLACE    LOGARITHMIC    FUNCTIONS. 


O          I 

L.  Sin. 

d. 

L.Tang. 

d. 

L.  Cotg 

L.  Cos. 

d. 

35    o 

10 
20 

3o 

4o 
5o 

9.7686 
9.7604 
9.7622 

9.7640 
9.7667 
9.7676 

18 
18 
18 

18 
18 
'7 

9.8462 

9.8479 
9.8606 

9.8533 
9.8669 
9.8686 

27 
27 
27 
26 

27 
27 
26 
27 
26 

27 
26 
26 
27 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 
26 

25 
26 
26 
25 

o.i548 
o.  1621 
0.1494 

o.  1467 
o.  i44i 
o.  i4i4 

ON  ON  ON  ON  ON  ON 

9134 
9126 
9116 

9107 
9098 
9089 

9 
9 

9 
9 
9 
9 

IO 

9 
9 

10 

9 

o    55 

5o 

4o 

3o 

20 
10 

36    o 

10 

20 

3o 

4o 
5o 

9 

9 
9 

9 
9 
9 

.7692 
.7710 
.7727 

•  7744 
.7761 

.7778 

9.86i3 
9.8639 
9.8666 

9.8692 

9.8718 
9.8746 

0.1387 
o.  i36i 
o.i334 

o.i3o8 
0.1282 
0.1266 

9.9080 
9.9070 
9.9061 

9.9062 
9.9042 
9.9033 

o    54 

5o 
4o 

3o 

20 
10 

37    o 

IO 

20 

3o 
4o 
5o 

9.7796 
9.7811 
9.7828 

9.7844 
9.7861 

9.7877 

I7 
16 

16 
'7 
16 

9.8771 
9.8797 
9.8824 

9.8860 
9.8876 
9.8902 

o.  1229 

O.  I2O3 

o.  1176 
o.  i  160 

O.  I  124 

0.1098 

9.9023 
9.9014 
9.9004 

9.8996 
9.8986 
9.8976 

9 

10 

9 

10 
10 

o    53 

5o 
4o 

3o 

20 
IO 

38    o 

10 

20 

3o 
4o 

5o 

000  000 

.7893 
.7910 
.7926 

.7941 
.7967 
.7973 

16 

15 
16 
16 

9.8928 
9.8964 
9.8980 

9.9006 
9.9032 
9.9068 

o.  1072 
o  .  i  o46 

O.  IO2O 

0.0994 

0.0968 
0.0942 

9.8966 
9.8966 

9.8946 

9.8935 
9.8926 
9.8916 

10 
10 
10 
10 
10 

o    52 

5o 

4o 

3o 

20 
IO 

39    o 

10 

20 

3o 
4o 
5o 

000  000 

.7989 
.8oo4 
.8020 

.8o35 
.8o5o 
.8066 

15 
16 

15 
15 
16 

9.9084 
9.9110 
9.9i35 

9.9161 
9.9187 
9.9212 

0.0916 
0.0890 

0.0866 
0.0839 

o.o8i3 
0.0788 

9.8906 
9.8896 

9.8884 

9.8874 
9.8864 
9-8853 

10 

II 

10 
10 

II 

o    51 

5o 
4o 

3o 
20 

IO 

40 

0 

9 

.8081 

15 

9.9238 

26 

0.0762 

9.8843 

o    50 

L 

.Cos. 

d. 

L.  Cotg. 

d. 

L.Tang. 

L.  Sin. 

d. 

'        0 

PP 

.1 

.2 

•3 

•4 
•5 
.6 

•  7 
.8 

•9 

36 

as 

18 

.2 

•3 
•4 

•7 

.8 

I? 

if 

15 

.1 

.2 

•3 

•4 

•5 
.6 

•7 

.8 

•9 

ii 

IO 

9 

2.6 
5-2 

7.8 

10.4 

13.0 

15-6 

18.2 

20.8 

23-4 

2-5 

5-o 
7-5 

IO.O 

12.5 
15.0 

17-5 

20.  o 

22.5 

1.8 
3-6 
5-4 

7.2 
9.0 
10.8 

12.6 

14.4 

' 

3-4 

6.8 
8-5 

10.2 

II-9 
13-6 

'5-3 

1.6 

tf 

6.4 
8.0 
9.6 

II.  2 

12.8 

14.4 

3-o 
4-5 

6.0 
7-5 
9.0 

10.5 
12:0 

i.i 

2.2 

3-3 
4-4 

1:1 
1.1 

1.0 

2.O 

4.0 
6.0 

7.0 
8.0 

9.0 

0.9 
1.8 
2.7 

3-6 
4-5 
5-4 

6-3 
7.9 
8.1 

FOUR-PLACE   LOGARITHMIC    FUNCTIONS. 


O         ' 

L.  Sin. 

d. 

L.Tang. 

d. 

L.  Cotgr. 

L.  Cos. 

d. 

40    o 

IO 
20 

3o 

4o 
5o 

9.8081 
9.8096 
9.8111 

9.8126 
9.8140 
9.8i55 

15 
15 
14 
15 
15 

9.9238 
9.9264 
9.9289 

9.93i5 
9.9341 
9.9366 

26 
25 
26 
26 

25 

26 

25 
26 

25 

26 

25 

25 
26 

25 
26 

25 

25 
26 

25 
25 
25 
25 
25 
25 
26 

25 
25 
25 
26 
25 

0.0762 
0.0736 
0.071  i 

o.o685 
0.0669 
o.o634 

9.8843 
9.8832 
9.8821 

9.8810 
9.8800 
9.8789 

ii 
ii 

10 

II 

o    50 

5o 
4o 

3o 
20 

IO 

41    o 

IO 

20 

3o 
4o 
5o 

9.8169 
9.8184 
9.8198 

9.8213 

9.8227 

9.8241 

15 
14 
15 
14 
14 

9.9392 
9.9417 
9.9443 

9.9468 
9.9494 
9.95i9 

0.0608 
o.o583 
o.o557 

o.o532 
o.o5o6 
o.o48i 

9 

9 
9 

9 
9 
9 

.8778 
.8767 
.8756 

.8745 
.8733 
.8722 

II 
II 
II 

12 
II 

o    49 

5o 
4o 

3o 

20 
IO 

42    o 

IO 

20 

3o 
4o 
5o 

9.8255 
9.8269 
9.8283 

9.8297 
9-83ii 
9.8324 

«4 
H 
J4 
14 
»3 

9.9544 
9.9570 
9.9595 

9.9621 
9.9646 
9.9671 

o.o456 
o.o43o 
o.o4o5 

0.0379 
o.o354 
0.0329 

9.8711 
9.8699 

9.8688 

9.8676 
9.8665 
9.8653 

12 
II 
12 
II 
12 

o   48 

5o 
4o 

3o 
20 

IO 

43    o 

IO 

20 

3o 
4o 
5o 

9.8338 
9.835i 
9.8365 

9.8378 
9.8391 
9-84o5 

'3 
'4 
»3 
13 
»4 

9.9697 
9.9722 
9.9747 

9.9772 
9.9798 
9.9823 

o.o3o3 
0.0278 
0.0253 

0.0228 

O.O2O2 
0.0177 

9 

9 
9 

9 
9 
9 

.864i 
.8629 
.8618 

.8606 
.8594 

.8582 

12 
II 
12 
12 
12 

o    47 

5o 

4o 

3o 

20 
10 

44    o 

IO 
20 

3o 
4o 
5o 

9.8418 
9-843i 
9-8444 

9.8457 
9.8469 
9.8482 

»3 
»3 
13 

12 

»3 

9.9848 
9.9874 
9.9899 

9  .  9924 
9.9949 
9.9975 

0.0152 

0.0126 

O.OIOI 

0.0076 
o.oo5i 

O.OO25 

9 
9 
9 

9 
9 
9 

.8569 

.8557 
.8545 

.8532 
.8520 
.85o7 

12 
12 
»3 
12 

13 

o    46 

5o 

4o 

3o 
20 

IO 

45    o 

9.8495 

0,0000 

o.oooo 

9 

.8495 

o   45 

L.  Cos. 

d. 

L.  Cotg. 

d. 

L.  Tang. 

L.  Sin. 

d. 

'          O 

PP 

.1 

.2 

•3 

•4 

:i 

.1 

•9 

26 

«5 

15 

M 

13 

12 

i 

.2 

•3 

•4 

:! 
:l 

•9 

IX 

IO 

2.6 
5-2 

78 

10.4 

130 

15-6 
182 

20.8 

23  4 

2-5 

5-o 
7-5 

10.0 

12  5 

150 

17  5 

20.0 

*  5 
3-o 

4-5 

60 

75 

9.0 

10.5 

I2.O 

13-S 

.2 

•  3 
•4 

:! 
;J 

•9 

a 

4-2 

5-6 
7-0 
8.4 

9.8 

H.2 

12.6 

a 

3-9 

5-2 

6-5 

7.8 

9.1 
104 
11.7 

1.2 

y 

4.8 

6.0 
7.2 

8-4 
9.6 

10.8 

I.I 

2.2 

3-3 
4.4 

1:1 

7-7 
8.8 

9-9 

1.0 
2.O 
3-0 

4-0 

5-° 
6.0 

7.0 
8.0 
9.0 

i48 


TABLE  VII 

FOUR-PLACE 

NATURAL    TRIGONOMETRIC 
FUNCTIONS 

TO   EVERY  TEN    MINUTES 


FOUR-PLACE    NATURAL   FUNCTIONS. 


O         ' 

Sin. 

d. 

Tang. 

d. 

Cotg. 

d. 

Cos. 

d. 

0    o 

10 

20 

3o 
4o 
5o 

o.oooo 
0.0029 
0.0068 

0.0087 
o.oi  16 
o.oi45 

29 
29 
29 
29 
29 
3» 
29 
29 
29 
29 
29 

o 
o 

o 

o 

0 

o 

.0000 
.0029 

.0068 

.0087 
.01  16 

.0145 

29 
29 
29 
29 
29 

3° 
29 
29 
29 
29 
29 
29 
29 
29 

3° 
29 
29 
29 
29 
29 

3° 
29 
29 
29 
30 
29 
29 
29 
30 
29 

infinit. 

343.7737 
171.8864 

114.6887 
86.9398 
68.7601 

it* 

818 
613 
477 
382 
312 
26c 

22C 

i8fi 
163 
H3 
126 

112 

IOC 

9c 
81 
7A 
6i 
6-. 
5y 
55 
4C 
4! 
4- 
3< 

i 

i 

0 
0 

.0000 
.0000 
.0000 

.0000 
•9999 
•9999 

o 
o 

o 
I 
o 

o  90 

5o 

4o 

3o 
20 

10 

1     o 

IO 
20 

3o 

4o 
5o 

0.0176 
0.0204 
0.0233 

0.0262 
0.0291 

O.O320 

0 

o 

o 

o 

0 

o 

.0176 
.  o£o4 
.0233 

.0262 
.0291 

032O 

67.2900 
49.  1039 
42.9641 

38.i885 
34.3678 
3i.24i6 

61 
98 
56 
07 
62 

o 
o 
o 

0 
0 
0 

.9998 
.9998 
•9997 

•9997 
.9996 
.9996 

0 

i 
o 

I 

I 
I 

2 

I 
I 

o  89 

5o 

4o 

3o 
20 

10 

2     o 

10 

20 

3o 
4o 
5o 

o.o349 
0.0378 

0.0407 

o.o436 
o.o465 
o.o494 

29 
29 
29 
29 
29 

o 

0 

o 

0 
0 
0 

.o349 
.o378 
.0407 

.o437 
.o466 
.0496 

28.6363 
26.43i6 
24.54i8 

22.9038 
21.4704 
20.2066 

bj 

47 
98 
80 

34 

48 

o 

0 
0 

0 

o 
o 

.9994 
.9993 
.9992 

•999° 
.9989 

.9988 

o  88 

5o 

4o 

3o 

20 
IO 

3    o 

10 

20 

3o 
4o 
5o 

0.0623 
0.0662 
o.o58i 

0.0610 
o  .o64o 
0.0669 

29 
29 
29 

30 
29 

0 

o 

0 

o 
o 

0 

.0624 
.o553 
.0682 

.0612 
.o64i 
.0670 

19.0811 
18.0760 
17.1693 

16.3499 
i5.6o48 
14.9244 

4b 
61 

57 
94 
5i 
04 

37 
40 
98 
07 
57 
43 
,61 

o 

0 

o 

0 
0 
0 

.9986 
.9986 
.9983 

.9981 
.9980 
.9978 

I 
2 
2 
I 
2 

o  87 

5o 
4o 

3o 
20 

10 

4    o 

10 
20 

3o 

4o 
5o" 

0.0698 
0.0727 
0.0766 

0.0786 
0.08  i  4 
o.o843 

29 
29 
29 
29 
29 
29 

0 
0 
0 

0 
0 
0 

.0699 
.0729 
.0768 

.0787 
.0816 
.o846 

i4.3oo7 
13.7267 
13.1969 

12.  7062 
I2.25o5 
11.8262 

0 
0 
0 

0 
0 

o 

.9976 
•9974 
.9971 

.9969 
.9967 
.9964 

2 

3 

2 
2 

3 

o  86 

5o 

4o 

3o 

20 
10 

5     o 

o. 

0872 

0 

.0876 

ii.43oi 

0 

.9962 

o  85 

Cos. 

d. 

Cotg. 

d. 

Tang. 

d. 

Sin. 

d. 

/          0 

PP 

.2 

•3 
•4 

J 

:i 

•9 

26053 

16380 

i  "45 

.i 

.2 

•  3 

•4 
•5 
.6 

•7 
.8 

•9 

8194 

6237 

4907 

.1 

.2 

•3 

•4 

•i 

:I 

•9 

3961 

30 

29 

2605 
5211 
7816 

10421 
13027 
15632 

18237 
20842 

23448 

1638 
3276 
4914 

6552 
8190 
9828 

11466 
13104 

14742 

1125 
2249 
3374 

4498 
5623 
6747 

7872 
8996 

IOI2I 

819.4 
1638.8 
2458-2 

3277-6 
4097.0 
4916.4 

5735-8 
6555.2 
r374-6 

623.7 
1247.4 
1871.1 

2494.8 

3"8.s 
3742.2 

4365-9 
4989.6 

490.7 
981.4 
1472.1 

1962.8 
2453-5 
2944.2 

3434-9 
3925.6 
4416. 

396-1 
792.2 
1188.3 

1584-4 
1980.5 
2376.6 

2772.7 
3168.8 
$564-9 

3-0 
6.0 
9.0 

12.0 
15-0 

18.0 

21.0 
24.0 

2.9 

5-8 

8-7 

ii.  6 
14-5 
17.4 

20.3 
23.2 
26.1 

160 


POUR-PLACE    NATURAL    FUNCTIONS. 


0        ' 

Sin. 

d. 

Tang. 

d. 

Cotg. 

d. 

Cos. 

d. 

5    o 

10 

2O 

0.0872 
0.0901 
0.0929 

29 
28 

o.o875 
0.0904 
0.0934 

29 

3° 

n.43oi 
u.o594 
io.7n9 

3707 

3475 

o 

0 
0 

.9962 

•9959 
•9957 

3 

2 

o    85 

5o 

4o 

3o 

0.0968 

29 
29 

0.0963 

^9 
29 

10.3854 

32^ 

5 

o 

.9954 

3 

•3 

3o 

4o 

0.0987 

0.0992 

io.o78o 

0 

.9951 

20 

5o 

o.  1016 

29 

O.  IO22 

3° 

9.7882 

0 

•  9948 

3 

10 

6    o 

10 

o.  io45 
o.  1074 

29 

o.  io5i 
o.  1080 

29 

9.5i44 
9.2553 

2738 
2591 

0 

0 

•  9945 
.9942 

3 

o    84 

5o 

20 

0. 

no3 

29 

O.IIIO 

3° 

9.oo98 

2455 

o 

•9939 

3 

4o 

3o 
4o 

o. 
o. 

I  I  32 

1161 

29 
29 

o.  i  189 

o.  1  169 

*y 

30 

8.7769 
8.5555 

2329 
2214 

0 

o 

.9936 
.9932 

3 
4 

3o 
20 

5o 

o. 

1190 

•'y 

0.1198 

^9 

8.345o 

0 

.9929 

3 

IO 

7    o 

IO 

20 

o. 
o. 
o. 

1219 
1248 
1276 

29 
28 

o.  1228 
o.  1267 
0.1287 

29 
30 

8.i443 
7.953o 
7.7704 

1913 
1826 

0 
0 

0 

.9926 
.9922 
.9918 

3 
4 

o    83 

5o 
4o 

3o 

o. 

i3o5 

29 

o.i3i7 

30 
29 

7.5958 

0 

.9914 

4 

3o 

4o 

o. 

i334 

o.i346 

7.4287 

0 

.9911 

20 

5o 

o. 

i363 

^9 

0.1376 

7.2687 

0 

.9907 

4 

IO 

8    o 

IO 

o. 

0. 

1392 
1421 

29 
29 

o.i4o5 
o.i435 

30 

7.ii54 

6.9682 

J533 
1472 

0 
0 

.9903 
.9899 

4 

o    82 

5o 

20 

o. 

1449 

o.i465 

3° 

6.8269 

I4I3 

0 

.9894 

5 

4o 

3o 

0. 

1478 

29 

o.i49 

3 

29 

6.69i2 

'357 

o 

.9890 

4 

3o 

4o 
5o 

o. 

0. 

1607 
i536 

29 

o.  1624 
o.i554 

30 

6.56o6 
6.4348 

1258 

0 

0 

.9886 
.9881 

5 

20 

IO 

9 

0 

o. 

1  564 

o.i584 

3° 

6.3i38 

,,68 

0 

.9877 

o    81 

IO 
20 

0. 
0. 

1693 
1622 

29 

o.i6i4 
o.  i644 

3° 

6.i97o 

6.o844 

1126 

0 
0 

.9872 
.9868 

4 

5o 

4o 

?K 

29 

10 

K 

5 

3o 

o. 

i65o 

0.1673 

3° 

5.9758 

0 

.9863 

3o 

4o 

o. 

i67g 

o.  1703 

5.87o8 

1050 

0 

.9868 

20 

5o 

0. 

1708 

29 

o.i733 

3° 

5.7694 

1014 

0 

.9853 

5 

IO 

10 

O 

o. 

i736 

o.i763 

5.6713 

901 

0 

.9848 

o    80 

Cos. 

d. 

Cotg. 

d. 

Tang. 

d. 

Sin. 

d. 

f        O 

PP 

2738 

1533 

981 

30 

29 

28 

5             4 

3 

tl 

273.8 

153-3 

98.1      .1 

3.0 

2.9 

2.8 

.! 

0.5          0.4 

o-3 

2 

547-6 

306.6 

196.2             .2 

6.0 

5-8 

5.6 

.2 

i.o          0.8 

0.6 

•3 

821.4 

459-9 

294-3          -3 

9.0 

8-7 

8.4 

•3 

1  5     i 

1.2 

0.9 

•4 

1095.2 

6'3-2 

392.4          .4 

12.  0 

n.6 

II.  2 

•4 

2.0 

1.6 

1.2 

•  5 

1369.0 

766.5 

490.5          .5 

15-0 

14-5 

14.0 

.5 

2-5 

2.O 

»-5 

.6 

1642.8 

919.8 

588.6         .6 

18.0 

J7-4 

16.8 

.6 

3-o 

2.4 

1.8 

.7 

1916.6 

1073.1 

686.7         -7 

21.0 

20.3 

19.6 

.7 

3-5 

2.8 

2.1 

.8 

2190.4 

1226.4 

784.8         .8 

24.0 

23.2 

22.4 

.8 

4.0 

3-2 

2.4 

2464.2 

T.7Q.7       882.0          .9 

27.0            26.1 

•9 

4.5          3.6          2.7 

POUR-PLACE   NATURAL   FUNCTIONS. 


0       ' 

Sin. 

d. 

Tang. 

d. 

Cotg. 

d. 

Cos. 

d. 

10    o 

IO 

20 

3o 
4o 
5o 

o. 

0. 

o. 

o. 
o. 
o. 

1735 

i765 

1794 
1822 

1861 

1880 

29 

29 
28 
29 

29 
28 
29 
28 
29 
28 
29 
28 

29 
28 
a8 

29 
28 

29 
28 
28 
28 
29 
28 
28 
28 

29 
28 
28 
28 

0.1763 

o.i793 
0.1823 

o.i853 
o.i883 
o.  1914 

3° 
3° 
3° 
3° 
31 
30 
30 
3° 
3i 
3° 
3° 
3i 
3° 
30 
31 
3° 
3i 

5.6713 
5.5764 
5.4845 

5.3955 
5.3o93 
5  .2257 

94 
9' 
8c 
8( 
8; 
81 
7* 
7< 
74 

r< 

7< 

6£ 
6f 
6< 
62 
61 
5c 
5* 
5< 
5! 
54 
52 
5i 
5C 
45 
4* 
4« 
45 
44 
43 

9 
9 

)0 

>2 

6 

0.9848 
0.9843 
0.9838 

o.9833 
0.9827 
0.9822 

5 
5 
5 
6 
5 

o  80 

5o 
4o 

3o 
20 

10 

11    o 

10 

20 

3o 
4o 
5o 

o. 
o. 

0. 

o. 
o. 
o. 

1908 
i937 
1966 

i994 

2O22 

2o5i 

0.1944 
o.  1974 

0.2OO4 

o.2o35 
o.2o65 
0.2095 

5.i446 
5.o658 
4.9894 

4-9i52 
4.843o 
4.7729 

8 
4 

2 
•2 
)j 

3 
4 
6 
9 
3 
7 

n 

0.9816 
0.9811 
0.9805 

0.9799 
0.9793 
0.9787 

5 
6 
6 
6 
6 

o  79 

5o 
4o 

3o 
20 

IO 

12   o 

IO 
20 

3o 
4o 
5o 

o. 
o. 
o. 

o. 
o. 
o. 

2079 
2IO8 

2i36 

2164 
2193 

2221 

0.2126 
o.2i56 
0.2186 

0.2217 
0.2247 
0.2278 

4.7046 
4.6382 
4.5736 

4.5i07 
4.4494 
4.3897 

0.9781 
0.9775 
0.9769 

0.9763 

0.9757 
0.9750 

6 
6 
6 
6 
7 

o  78 

5o 
4o 

3o 
20 

10 

13  o 

IO 

20 

3o 
4o 
5o 

o. 

0. 
0. 

o. 
o. 
o. 

225O 

2278 

23o6 

2334 
2363 
2391 

0.2309 
0.2339 
0.2370 

0.2401 
0.2432 
0.2462 

30 
31 
3i 
3i 
30 

4.33i5 

4.2747 
4.2193 

4.i653 
4.  1126 
4.0611 

8 

4 

0 

7 

5 
3 
i 
i 
9 
9 
8 

9 

0.9744 
0.9737 
0.9730 

0.9724 
0.9717 
0.9710 

7 
7 
6 

7 
7 
7 
7 
7 
8 

7 

7 
g 

o  77 

5o 
4o 

3o 
20 

IO 

14  o 

IO 

20 

3o 
4o 
5o 

o. 
o. 
o. 

o. 

o. 

0. 

2419 

244? 
2476 

2604 

2532 

256o 

0.2493 
0.2524 

0.2555 

0.2586 
0.2617 
0.2648 

3* 
3i 
3i 
3» 
3» 

4.0108 
3.9617 
3.9i36 

3.8667 
3.8208 
3.7760 

0.9703 
0.9696 
0.9689 

0.9681 
0.9674 
0.9667 

o  76 

5o 
4o 

3o 

20 
IO 

15 

O 

o. 

2588 

0 

.2679 

3.732i 

0 

.9659 

o  75 

Cos. 

d. 

Cotg. 

d. 

Tang. 

d. 

Sin. 

d. 

'     o 

PP 

i 

2 

3 
4 

I 

I 

742 

448 

31 

.1 

.2 

•3 

•4 
-5 
.6 

:i 

30 

29 

28 

.1 

.2 

•3 

•4 

•5 
.6 

•  7 
.8 

7 

6 

5 

74.2 
148.4 

222.6 

296.8 
371-0 

445-2 

S'9-4 
593-6 

44-8 
89.6 
134-4 

179.2 
224.0 
268.8 

313-6 
358.4 
403.2 

t: 

9-3 
12.4 

;i:I 

21.7 
24.8 

3-o 
6.0 
9.0 

12.0 
15-0 
18.0 

21.0 

24.0 

*« 

8.7 

ii.  6 
M-5 
17.4 

20.3 

2.8 

5-6 
8.4 

II.  2 
14.0 

16.8 

19.6 
22.4 

0.7 
1.4 

2.1 
2.8 

3-5 

4-2 

4.9 

r 

0.6 

1.2 

1.8 

2-4 

H 

ti 

5-4 

o-5 

I.O 

i-5 

2.0 
2-5 

3-o 

3-5 
4.0 

4-5 

152 


POUR-PLACE   NATURAL    FUNCTIONS. 


O         ' 

Sin. 

d. 

Tang. 

d. 

Cotg. 

d. 

Cos. 

d. 

15    o 

10 

20 

3o 
4o 
5o 

o. 

0. 
0. 

o. 
o. 

0. 

2588 
2616 
2644 

2672 
2700 
2728 

28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
27 
28 
28 
27 
28 
28 

27 
28 
28 
27 

0.2679 
0.271  i 
0.2742 

0.2773 
o.28o5 
0.2836 

32 
3» 

32 

32 
32 
3* 
32 
32 
3» 
32 
32 
32 
32 
32 

3.7321 
3.6891 
3.6470 

3.  6059 
3.5656 
3.526i 

4 
4 
4 
4 
3 
3 
3 
3 

3 
3 
3 
3 
3 
3 
3 
3 
3< 
3 

2 
2 
2 
2 
2 
2 
2< 
2< 
2. 
2. 
2. 

3° 

:i 
ii 
33 
M 

37 

79 
71 
^5 
37 
50 

0.9659 
0.9652 
0.9644 

0.9636 
0.9628 
0.9621 

7 

8 
8 
8 
7 

o    75 

5o 
4o 

3o 
20 

IO 

16    o 

IO 

20 

3o 

4o 
5o 

o. 
o. 
o. 

o. 
o. 
o. 

2756 
2784 
2812 

2840 
2868 
2896 

0.2867 
0.2899 
0.2931 

0.2962 
0.2994 
o.3o26 

3.4874 
3.4495 
3.4i24 

3.3759 
3.34o2 
3.3o52 

0.9613 
0.9605 
0.9596 

0.9588 
0.9580 

0.9572 

8 

9 
8 
8 
8 

o    74 

5o 

4o 

3o 
20 

IO 

17    o 

IO 

20 

3o 

4o 
5o 

0.2924 
0.2952 

0.2979 

0.3007 
o.3o35 
o.3o62 

0.3089 

0.  3l2I 

o.3i53 

o.3i85 
0.3217 

3.2709 
3.237i 
3.2o4i 

3.1716 
3.i397 

3.io84 

13 
38 
5° 

25 
9 
3 

0.9563 
o.9555 
o.9546 

o.9537 
o.9528 

O.952O 

8 
9 
9 
9 

8 

o    73 

5o 
4o 

3o 
20 

IO 

18    o 

IO 
20 

3o 
4o 
5o 

o. 
o. 
o. 

o. 
o. 
o. 

3090 
3n8 
3i45 

3i73 

3201 
3228 

0.3249 
o.328i 
o.33i4 

0.3346 
o.3378 
o.34n 

32 

33 
32 
32 
33 

3.0777 
3.o475 
3.0178 

2.9887 
2.9600 
2.93i9 

Jl 

m 

17 

H 

37 
h 

o.95n 
o.95o2 
o.9492 

o.9483 
o.9474 
o.9465 

9 

IO 

9 
9 
9 

o    72 

5o 
4o 

3o 
20 

IO 

19    o 

IO 
20 

3o 
4o 
5o 

o. 
o. 
o. 

o. 
o. 
o. 

3256 
3283 
33u 

3338 
3365 
3393 

27 
28 
27 

27 
28 

0.3443 
0.3476 
o.35o8 

o.354i 
0.3574 
0.3607 

33 

33 
33 
33 

2.9042 
2.877o 

2.8502 

2.8239 

2.  -7980 
2.  7725 

1 

2 

,8 
>3 
9 
5 

o.9455 
o.9446 
o.9436 

o.9426 
o.94i7 
o.94o7 

9 

10 
10 

9 

IO 

o   71 

5o 
4o 

3o 
20 

IO 

20 

0 

o. 

3420 

27 

o.364o 

2.7475 

0 

o.9397 

o    70 

Cos. 

d. 

Cotg 

d. 

Tang. 

d 

• 

Sin. 

d. 

'        O 

PP 

.1 

.2 

•3 
•4 

:J 

•9 

255 

33 

32 

31 

28 

27 

10 

9 

8 

25-5 
51.0 
76.5 

102.0 

127.5 
i53-o 

178-5 
204.0 

1:1 

9-9 

13.2 
16.5 
19.8 

23.1 
26.4 
29-7 

3.2 

6.4               .2 

9.6               .3 

12.8               .4 

16.0          .5 
19.2           .6 

22.4           .7 
25.6          .8 

11 

9-3 

12.4 

S3 
£i 

2.8 

tf 

II.  2 
14-0 

16.8 

19.6 
22.4 

2-7 

K 

10.8 

'3-5 
16.2 

18.9 

21.6 

24-3 

.1            I.O 
.2            2.0 

•3        3-° 

.4        4.0 
•5         5'° 
.6        6.0 

•7         7-° 
.8         8.0 
.9         9.0 

a 

2-7 

3-6 
4-5 
5-4 

6-3 

K 

0.8 
1.6 
2.4 

3-2 

4.0 
48 

5-6 
6.4 

7-2 

i53 


FOUR-PLACE    NATURAL    FUNCTIONS. 


O         ' 

Sin. 

d. 

Tang. 

d. 

Cotg. 

d. 

Cos. 

d. 

20    o 

IO 
20 

3o 
4o 
5o 

o. 
o. 

0. 

o. 

0. 

o. 

3420 
3448 
3475 

35o2 
3529 
3557 

28 
27 
27 
27 
28 
27 
27 
27 
27 
27 
27 

o.364o 
0.3673 
0.3706 

0.3739 

0.3772 

o.38o5 

33 

33 
33 
33 
33 

2.7475 
2.7228 
2.6985 

2.6746 
2.65n 
2.6279 

247 
243 
239 
235 
232 

228 

o.9397 
0.9387 
o.9377 

o.9367 
o.9356 
o.9346 

IO 
IO 
10 

II 

IO 

o    70 

5o 

4o 

3o 
20 

IO 

21    o 

IO 
20 

3o 
4o 
5o 

o. 
o. 

0. 

o. 

0. 
0. 

3584 
36n 
3638 

3665 
3692 
37i9 

0 
0 
0 

0 
0 
0 

.3839 
.3872 
.3906 

.3939 
.3973 
.4oo6 

33 
34 
33 
34 
33 

2.6o5i 
2.5826 
2.56o5 

2.5386 
2.5172 
2.4960 

225 

221 
2I9 
214 
212 

o.9336 
o.9325 

o.93o4 
o.9293 
0.9283 

II 

IO 

II 
II 

IO 

o    69 

5o 
4o 

3o 
20 

IO 

22    o 

IO 

20 

3o 

4o 

5o 

o. 
o. 
o. 

o. 
o. 
o. 

3746 
3773 
38oo 

3827 
3854 
388i 

27 
27 
27 

27 
27 

0 
0 
0 

0 
0 
0 

.4o4o 
•  4o74 
.4108 

'•4i76 
.4210 

34 
34 
34 
34 
34 

2.4545 
2.4342 

2.3945 
2.3750 

206 
203 
2OO 
I97 
195 

0.9261 
0.9250 

0.9239 
0.9228 
0.9216 

II 
II 
II 
II 

12 

o    68 

5o 
4o 

3o 
20 

IO 

23    o 

10 
20 

3o 
4o 
5o 

o.39o7 
o.3934 
o.396i 

o.3987 
o.4oi4 
o.4o4i 

27 
27 
26 
27 
27 
26 
27 
26 
27 
26 
27 

0 
0 
0 

0 
0 
0 

.4245 

•4279 

.43i4 

.4348 
.4383 
.44i7 

35 
34 
35 
34 
35 
34 
35 
35 
35 
35 
35 
36 
35 

2.3559 
2.3369 
2.3i83 

2.2998 
2.2817 
2.2637 

I90 

1  86 
185 
181 
1  80 
177 
174 
i73 
170 
168 
1  66 
i6<i 

0.9205 
o  .  9  i  94 
0.9182 

o.9i7i 
0.9159 

o.9i47 

II 
12 

II 
12 
12 

o    67 

5o 
4o 

3o 
20 

IO 

24    o 

10 
20 

3o 
4o 
5o 

o. 
o. 
o. 

0. 

o. 
o. 

4o67 
4094 
4120 

4i47 
4173 
4200 

0 
0 
0 

0 
0 

o 

.4452 
.4487 

.4522 

.4557 
.4592 
.4628 

2.2460 
2.2286 

2.2II3 

2.i943 

2.1775 
2.  1609 

0.9135 
0.9124 
0.9112 

o.9ioo 

o.9o88 
o.9o75 

II 
12 

12 
12 

13 
12 

o    66 

5o 
4o 

3o 
20 

IO 

25 

0 

o. 

4226 

0 

.4663 

2.i445 

0 

.9o63 

o    65 

Cos. 

d. 

Cotg. 

d. 

Tang. 

d. 

Sin. 

d. 

'         O 

PP 

.1 

.2 

•3 

•4 
•5 
.6 

177 

35 

34 

.2 

•3 

•4 
•  5 
.6 

•  7 
.8 

•9 

33 

27 

26 

.1 

.2 

•3 

•4 
•5 
.6 

12 

ii 

IO 

17.7 
35-4 

70.8 
88.5 
106.2 

123.9 
141.6 

3-5 
7.0 
10.5 

14.0 

21.0 

24-5 
28.0 
3z-5 

U 

10.2 

13-6 
I7.0 
20.4 

23.8 
27.2 

1:1 

9-9 

13.2 
16.5 
19.8 

III 

2.7 

5-4 
8.1 

10.8 
'3-5 
16.2 

18.9 

21.6 

24-3 

2.6 

10.4 

13.0 

15-6 

18.2 

20.8 

23-4 

1.2 

2.4 

3-6 

6.0 
7.2 

8.4 
9.6 
10.8 

2.2 

3-3 

4-4 

ii 

1.1 

1.0 
2.O 

3-o 

4.0 
5-° 
6.0 

7.0 
8.0 

1 54 


FOUR-PLACE   NATURAL    FUNCTIONS. 


O         ' 

Sin. 

d. 

Tang. 

d. 

Cotg. 

d. 

Cos. 

d. 

25 

0 

o. 

4226 

0.4663 

06 

2.i445 

162 

0.9063 

o   65 

IO 

20 

o. 
o. 

4253 

4279 

26 

0.4699 
0.4734 

35 
36 

2.1283 
2.II23 

160 
158 

0.9051 
0.9038 

13 

12 

5o 
4o 

3o 

o. 

43o5 

26 

0.4770 

06 

2.0965 

1*6 

0.9026 

3o 

4o 

o. 

433i 

o.48o6 

2.0809 

0.9013 

20 

5o 

0. 

4358 

o.484i 

35 

2  o655 

J54 

0.9001 

IO 

26 

0 

0. 

4384 

26 

0.4877 

36 

2.o5o3 

0.8988 

o    64 

10 

o. 

44io 

0.4913 

2.o353 

0.8975 

5o 

20 

0. 

4436 

26 

0.495 

0 

37 

2.O2O4 

149 

0.8962 

IJ 

4o 

fh 

36 

i 

7 

JO 

3o 

o. 

4462 

26 

0.4986 

16 

2.0057 

0.8949 

3o 

4o 

o. 

4488 

O.5O22 

.9912 

0.8936 

20 

5o 

o. 

45i4 

26 

o.5o5 

9 

37 

.9768 

144 

0.8923 

M 

IO 

27 

0 

o. 

454o 

26 

o.Sog 

5 

.9626 

0.8910 

o    63 

10 

o. 

4566 

o.5i32 

.9486 

0.8897 

5o 

20 

o. 

4592 

26 

0.5169 

37 

.9347 

1 

59 

0.8884 

1J 

4o 

3o 

4o 

0. 

o. 

46i7 
4643 

25 

26 

0.5206 
0.5243 

37 
37 

.9210 
.9074 

136 

0.8870 
0.8857 

13 

3o 

20 

5o 

0.4669 

26 

0.5280 

37 

.8940 

134 

0.8843 

M 

IO 

28 

0 

o. 

4695 

o.53i7 

.8807 

X33 

0.8829 

o    62 

10 

o. 

4720 

0.5354 

.8676 

0.8816 

5o 

20 

o. 

4746 

20 

0.5392 

3^ 

.8546 

130 

0.8802 

14 

4o 

3o 

0. 

4772 

26 

o.543o 

38 

.84i8 

128 

0.8788 

14 

3o 

4o 
5o 

o. 
o. 

4797 
4823 

36 

o.  5467 
o.55o5 

38 

.8291 
.8i65 

126 

0.8774 
0.8760 

20 
IO 

29 

0 

o. 

4848 

0.5543 

38 

i.8o4o 

I25 

0.8746 

o   61 

IO 

o. 

4874 

o.558i 

S8 

1.7917 

123 

0.8732 

14 

5o 

20 

o. 

4899 

25 

0.5619 

3» 

1.7796 

121 

0.8718 

M 

4o 

3o 

4o 

o. 
o. 

4924 

25 
26 

0.5658 
0.5696 

39 
38 

1.7675 
1.7556 

121 

0.8704 
0.8689 

14 
15 

3o 
20 

5o 

o. 

4975 

•3 

0.5735 

39 

i.7437 

o.8675 

'4 

IO 

30 

0 

o. 

5ooo 

0.5774 

1.7321 

0.8660 

o    60 

Cos. 

d. 

Cotg. 

d. 

Tang. 

d. 

Sin. 

d. 

>      o 

PP 

149 

I3i 

39 

38 

37 

36 

25 

M 

13 

.  I 

14.9 

2Q  8 

26  2 

3-9           -i 

78                 2 

3-8 

3-7 

3-6 

.1 

2.5 

LI 

26 

•3 

44-7 

39-3 

"•7           -3 

11.4 

II.  I 

10.8 

•3 

7-5 

4-2 

3-9 

•4 

59-6 

52.4 

15.6           .4 

15.2 

14-8 

14.4 

•4 

10.  0 

5-6 

5-2 

.5 

74-5 

65-5 

»9-5           -5 

19.0 

i8.S 

18.0 

•5 

12.5 

7.0 

6-5 

.6 

89.4 

78.6 

23-4           -6 

22.8 

22.2 

21.6 

.6 

15-0 

8.4 

7.8 

•7 

104.3 

91.7 

27-3           -7 

26.6 

25.9 

25-2 

.7 

I7-5 

9.8 

9.1 

.8 

119.2 

104.8 

31.2           .8 

30.4 

29.6 

28.8 

.8 

20.0 

II.  2 

10.4 

117.9         35-i             -9 

32-4 

22.5 

12.6        11.7 

i55 


FOUR-PLACE    NATURAL    FUNCTIONS. 


o 

' 

s 

in. 

d. 

1 

rang 

. 

d. 

Cot£ 

r. 

d 

. 

Cos. 

d. 

30 

i 

£ 

i 
| 

0 
0 
0 

0 

^o 

0 

o. 
o. 
o. 

o. 
o. 
o. 

5ooo 
5o25 
5o5o 

5o75 
5ioo 
5i25 

25 
25 
25 
25 
25 

0 
0 
0 

0 
0 

0 

.577 

.58i 
.585 

.589 
.593 
.596 

i 
2 
I 

D 
3 
I 

38 

39 
39 
40 

39 

.73 

I.72< 
.70( 

.69r 

.68( 
.67, 

21 

D5 
?° 

77 
34 
)3 

ii 

ii 
U 

ii 
ii 

6 

5 
3 
3 

C 

0 
0 

0 
0 

o 

.8660 

.8646 
.863i 

.8616 
.8601 

.8587 

14 
15 
15 
15 

J4 

o    60 

5o 
4o 

3o 

20 
IO 

31 

i 

£ 

i 
\ 

0 
0 
0 

0 

o 

0 

o. 
o. 
o. 

o. 
o. 
o. 

5i5o 
5i75 

52OO 
5225 

525o 
5275 

25 
25 
25 
25 
'25 

0 
0 

o 

o 

o 

0 

.600 
.6o4i 
.608 

.612! 
.616! 
.620! 

) 

3 

3 
3 
3 

39 
40 
40 
40 
40 

.66. 

.65: 

.645 

.63 
.62: 
.6ic 

i3 
M 

26 

9 

2 

>7 

1C 
1C 
10 

10 
10 

9 

8 

7 
7 

5 

0 
0 

o 

o 
o 

0 

.8572 
.8557 
.8542 

.8526 
.85n 
.8496 

i5 
15 
16 
15 

15 
16 

o    59 

5o 
4o 

3o 
SO 

IX) 

32 

i 

£ 
c 

/! 
i 

0 
0 

o 

0 
0 

0 

o. 
o. 
o. 

o. 
o. 
o. 

5299 
5324 
5348 

5373 
5398 
5422 

25 
24 
25 
25 
24 

0 
0 
0 

0 

o 

0 

.6241 
.628* 
.633* 

.637 
.64i! 

.645 

? 

? 

5 

[ 
2 

3 

40 
41 
41 
4i 
4i 

.6o( 
.59c 

.57c 

.56c 
.55c 
.54c 

>3 

)0 

>8 

>7 

>7 
^7 

10 
10 
1C 
10 
IO 

3 

2 

I 
0 
O 

o 
o 
o 

o 
o 
o 

.848o 
.8465 
.845o 

.8434 
.84i8 
.84o3 

15 
IS 

16 
16 

15 
ifi 

o    58 

5o 

4o 

3o 

20 
10 

33 

i 

£ 

4 

c 

C) 
0 
0 

0 
0 
0 

o. 
o. 
o. 

o. 
o. 
o. 

5446 
5471 
5495 

55i9 

5544 
5568 

25 
24 
24 
25 
24 

0 
0 
0 

0 
0 
0 

.649^ 
.6531 
.657 

.66i< 
.666 
.670 

i 

7 

? 

[ 
5 

42 
4i 
42 
42 
42 

.53c 
.53c 

.52( 

.5i( 
,5oi 

•  49 

>9 

>i 

>4 

>8 
3 

9 

y 

9 
9 
9 
9 
9 

8 

7 

6 

5 

4 

o 
o 

0 

o 
o 

0 

.8387 
.837i 
.8355 

.8339 
.8323 
.83o7 

16 
16 
16 
16 
16 

o    57 

5o 

4o 

3o 
20 

10 

34 

i 

£ 

C 

2 
f. 

0 
0 

o 

0 

o 

0 

o. 
o. 
o. 

o. 
o. 
o. 

5592 
56i6 
564o 

5664 
5688 
5712 

24 
24 
24 
24 
24 
24 

0 
0 
0 

0 
0 
0 

.674 
.678 
.683< 

.687 
.69i( 
.695« 

7 

5 

5 

I 

42 
43 
43 
43 
43 

.48i 

.47: 
.4& 

.45J 
•  44( 
.43- 

56 

*3 
il 

)0 
)0 
JO 

y 

9 
9 
9 
9 
9 

3 

2 
I 
0 
O 

o 
o 
o 

o 
o 

0 

.829o 

.8274 

.8258 

.8241 
.8225 
.8208 

J7 
16 
16 
*7 
16 

17 
16 

o    56 

5o 
4o 

3o 
20 

IO 

35 

0 

o. 

5736 

24 

0 

.700 

I 

43 

1.42* 

h 

0 

.8l92 

o    55 

c 

OS. 

d. 

( 

:otg. 

d. 

Tanj 

?• 

d 

• 

Sin. 

d. 

'          0 

PP 

13 

42 

41 

40 

25 

2 

t 

17 

16 

15 

.1 

.2 

•3 
•4 

:l 

:i 

I 

2 
2 

3 

3 

tl 

2.9 
7.2 

;:i 

0.1 

ti 

t; 

12.6 

16.8 

21.  0 
25-2 

29.4 

33-6 

t 

12. 

16. 

20. 
24. 

28. 

I 

2 

3 
4 

! 

.1 

.2 

•3 
•4 

' 

i 
i 

i 

4.0 

8.0 

2.O 

6.0 

0.0 

4.0 

8.0 

J2.0 

(6.0 

2.5 

5-o 
7-5 

IO.O 

12.5 
15-0 

17-5 
20.  o 

2 

4 
7 

9 

12 
H 

16 
"9 

21 

5 

2 

6 
o 

4 

8 

.2 

.6 

I 

2 

3 

4 
5 
6 

78 

*-7 
3-4 
5-i 

6.8 
8-5 

10.2 

II.9 
I3.6 

1.6 

43:I 

6.4 
8.0 
9.6 

II.  2 
12.8 
14.4 

1.5 

3-° 
4-5 

6.0 
75 
9.0 

10.5 

12.0 
13-5 

1 56 


POUR  PLACE    NATURAL    FUNCTIONS. 


o      / 

Sin. 

d. 

Tang. 

d. 

Cotg. 

d. 

Cos. 

d. 

35    o 

10 

20 

3o 
4o 
5o 

o 
o 
o 

0. 

o. 
o 

5736 
6760 
5783 

58oy 
583i 

5854 

24 
23 
24 
24 
23 

0.7002 

o.7o46 
o.7o89 

o.7i33 

o.7i77 

O.722I 

44 
43 
44 
44 
44 
44 
45 
45 
45 
45 
45 
46 
45 
46 
46 
47 
46 

.4281 

.4i93 
•  4io6 

.4oi9 
.3934 

,3848 

88 
87 
87 
85 
86 
84 
84 
83 
83 
82 
Si 

0.8l92 

o.8i75 
o.8i58 

o.8i4i 
0.8124 
o.8io7 

*7 
i7 
17 
J7 
17 

o    55 

5o 

4o 

3o 

20 
IO 

36    o 

10 

20 

3o 

4o 
5o 

o. 

0. 

o. 

o. 
o. 
o. 

5878 
5901 
5925 

5948 
6972 
5995 

23 
24 
23 
24 
23 

O.7265 

o.73io 
o.7355 

o.74oo 
o.7445 
o.749o 

.3764 
.368o 
.3597 

.35i4 
.3432 
.335i 

o.8o9o 
o.8o73 
o.8o56 

o.8o39 
0.8021 
o.8oo4 

*7 
17 
*7 
18 

17 

o    54 

5o 

4o 

3o 
20 

10 

37    o 

10 

20 

3o 

4o 
5o 

o. 
o. 
o. 

o. 
o. 
o. 

6018 
6o4i 
6o65 

6088 
61  1  1 
6i34 

23 
24 
23 
23 
23 

o.7536 
o.758i 

O.7627 

o.7673 

O.772O 

o.7766 

.3270 

.3i9o 
.3iu 

.3o32 
.2954 
.2876 

80 
79 
79 
78 
78 

o.7986 
o.7969 
o.795i 

o.7934 
o.79i6 

o.7898 

17 
18 

17 

.8 
18 

o    53 

5o 

4o 

3o 
20 

10 

38    o 

10 

20 

3o 

4o 
5o 

o. 
o. 
o. 

o. 
o. 

0. 

6i57 
6180 
6202 

6225 
6248 

627I 

23 

22 

23 
23 
23 

o.78i3 
o.786o 
o.79o7 

o.7954 
0.8002 
o.8o5o 

47 
47 
47 
47 
48 
48 

.2799 

.2723 

.2647 

.2572 
•  2497 
.2423 

77 
76 
76 
75 
75 
74 

o.788o 
o.7862 
o.7844 

o.7826 
o.78o8 
o.779o 

18 
18 
18 
18 
18 

o    52 

5o 
4o 

3o 

20 
IO 

39   o 

10 

20 

3o 
4o 
5o 

o. 

0. 
0. 

o. 

0. 
0. 

6293 
63i6 
6338 

636i 
6383 
64o6 

23 
22 

23 
22 

23 

o.8o98 
o.8i46 
o.8i95 

0.8243 

O.8292 

0.8342 

4* 
48 

49 
48 

49 
So 

.2349 

.2276 
.2203 

.2l3l 

.2o59 

.1988 

74 

73 
73 
72 
72 
7i 

o.777i 
o.7753 
o.7735 

o.77I6 
o.7698 
o.7679 

18 
18 

*9 
18 

19 

o    51 

5o 
4o 

3o 

20 
IO 

40 

0 

0. 

6428 

o.839i 

49 

i.i9i8 

70 

o.766o 

'9 

o    50 

Cos. 

d. 

Cotg. 

d. 

Tang. 

d. 

Sin. 

d. 

t         O 

PP 

.1 

.2 

•3 

•4 

•  5 
.6 

:l 

48 

47 

46 

45 

44 

23 

.1 

.2 

•3 

•4 

•5 
.6 

:! 

22 

19 

18 

4.8 

9.6 
14.4 

19.2 
24.0 
28.8 

tf 

43-2 

4-7 
9.4 
14.1 

18.8 

2238:2 
32.9 

37-6 
42.3 

4.6 

9.2                  .2 

13-8            -3 

18.4            .4 

23-°            -5 
27.0            .6 

32-2            .7 
36.8            .8 
41.4            .9 

4-5 
9.0 
13-5 

18.0 
22.5 
27.0 

3'  -5 
36.0 
40.5 

44 
8.8 
»3-2 

17.6 

22.0 

26.4 

30.8 

35-2 

2.3 
4.6 
6.9 

9.2 
"•5 
13-8 

16.1 
18.4 

2.2 

4-4 
6.6 

8.8 

II.  0 

13.2 

3 

19.8 

i.9 
3-8 
5.7 

7-6 
9-5 
11.4 

13-3 
15-2 

17.1 

1.8 
3-6 
5-4 

7-2 
9.0 
10.8 

12.6 

14.4 

16.2 

i57 


POUR-PLACE  NATURAL   FUNCTIONS. 


O         ' 

Sin. 

d. 

Tang. 

d. 

Cotg. 

d. 

Cos. 

d. 

40    o 

10 

20 

3o 

4o 
5o 

0.6428 
o.645o 
0.6472 

0.6494 
0.6617 
o.6539 

22 
22 
22 

23 

22 

0.8391 

o.844i 
0.8491 

o.854i 
0.8691 
0.8642 

50 

5° 
5° 
So 
51 
Si 
51 
52 
51 
52 
53 
52 
53 
53 
53 
54 
54 
54 
55 
55 
55 
55 
56 
c.6 

1.1918 

1.1847 
1.1778 

1.1708 
i.i64o 
i  .  1671 

7* 
69 
70 

68 
69 

o.  7660 
0.7642 

0.7623 
0.7604 

0.7686 
0.7566 

18 
*9 
»9 
19 
J9 

o   50 

5o 
4o 

3o 

20 
10 

41    o 

10 
20 

3o 
4o 
5o 

o. 
o. 
o. 

o. 
o. 
o. 

656i 
6583 
66o4 

6626 

6648 
6670 

22 
21 
22 
22 
22 

0.8693 

o.8744 
0.8796 

0.8847 
0.8899 
0.8962 

i.i5o4 
i.i436 
1.1369 

i.i3o3 
1.1237 
1.1171 

67 
68 
67 
66 
66 
66 

0.7647 
0.7628 

0.7609 

0.7490 
0.7470 
0.7461 

«9 

19 

'9 
19 

20 
19 

o    49 

5o 
4o 

3o 
20 

IO 

42    o 

10 

20 

3o 
4o 
5o 

o. 
o. 

0. 

o. 

o. 
o. 

6691 
67i3 
6734 

6756 
6777 
6799 

22 
21 
22 
21 
22 

0.9004 
0.9067 
0.9110 

0.9163 
0.9217 
0.9271 

i  .  i  106 
i  .io4i 
1.0977 

i  .0913 
i.o85o 
1.0786 

65 
65 
64 
64 
63 
64 

o.743i 
0.7412 
0.7392 

o.7373 
o.7353 
o.7333 

»9 

20 

19 
2O 
20 

o   48 

5o 

4o 

3o 

20 
10 

43    o 

10 
20 

3o 
4o 
5o 

o. 
o. 
o. 

o. 
o. 
o. 

6820 
684i 
6862 

6884 
6906 
6926 

21 
21 
22 
21 
21 

0.9326 
0.9380 
0.9435 

0.9490 
0.9645 
0.9601 

i  .0724 
i  .0661 
i  .0699 

i.o538 
1.0477 
i  .  o4  i  6 

63 
62 
61 
61 
61 

o.73i4 
o.7294 

0.7274 

o.7254 
0.7234 
0.7214 

'9 

20 
20 
2O 
20 
2O 

o    47 

5o 
4o 

3o 

20 
IO 

44   o 

IO 
20 

3o 
4o 

5o 

0.6947 
0.6967 

0.6988 

0.7009 
0.7030 
0.7060 

20 
21 
21 
ZI 
20 
21 

0.9667 
0.9713 
0.9770 

0.9827 
0.9884 
0.9942 

56 
57 
57 
57 
58 

i.o355 
1.0296 
1.0235 

i  .0176 
1.0117 
i.  0068 

60 
60 
59 
59 
59 

0.7193 
o.7i73 
o.7i53 

o.7i33 

O.7II2 
O.7O92 

20 
20 
20 
21 
20 

o   46 

5o 
4o 

3o 

20 
10 

45 

0 

o. 

7071 

i  .0000 

i  .0000 

O.7O7I 

o   45 

Cos. 

d. 

Cotg. 

d. 

Tang. 

d. 

Sin. 

d. 

'      o 

PP 

.1 

.2 

•3 

•4 

j 

:i 

•9 

57 

55 

54 

53 

51 

22                              21 

20 

19 

5-7 
11.4 
17.1 

22.8 

28.5 

34-2 

39-9 
45.6 

5  1  •  .1 

5-5 

II.  O 

16.5 

22.0 
27-5 

33-o 

38.5 
44.0 
49-5 

5-4          •' 

10.8               .2 

16.2          .3 

21.6         .4 
27.0        .5 
32.4          -6 

37-8          -7 
43-2          .8 
48.6          .9 

5-3 

10.  0 

iS-9 

21.2 

26.5 
31.8 

37-i 
42.4 
47-7 

5-i 

10.2 
»5-3 

20-4 
25-5 
30-6 

35-7 
40.8 

45-9 

2.2                .1             2.1 
4.4                .2             4.2 

6.6          .3         6.3 

8.8           .4         8.4 
ii.  o           .5        10.5 
13.2           .6       12.6 

*5-4           -7       14-7 
17.6           .8       16.8 
19.8           .9       18.9 

2.0 
4-0 

6.0 
8.0 

IO.O 
12.  0 

14.0 
IO.O 

18.0 

3 

5-7 

7-6 
9-5 
11.4 

i3-3 
15.2 
17.1 

i58 


TABLE   VIII. 
SQUARES   AND   SQUARE   ROOTS   OF  NUMBERS. 

SQUARES  OF  INTEGERS  FROM  10  TO  100. 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

mmmumum 

IO 

MMHMM^ 

100 

^^^^••M 

121 

1  44 

mmmmm^^ 

169 

196 

226 

256 

289 

324 

36i 

20 

4oo 

44  1 

484 

629 

576 

625 

676 

729 

784 

84  1 

3o 

900 

961 

1024 

1089 

n56 

1225 

1296 

1  369 

1  444 

l52I 

4o 

1600 

1681 

1764 

1849 

i936 

2025 

2116 

2209 

23o4 

2401 

5o 

2600 

2601 

2704 

2809 

2916 

3o25 

3i36 

3249 

3364 

348  1 

60 

36oo 

8721 

3844 

3969 

4096 

4225 

4356 

4489 

4624 

4761 

70 

4900 

5o4i 

5i84 

5329 

5476 

56a5 

5776 

5929 

6o84 

6241 

80 

64oo 

656i 

6724 

6889 

7066 

7225 

?396 

7569 

7744 

7921 

90 

8100 

8281 

8464 

8649 

8836 

9025 

9216 

9409 

9604 

9801 

SQUARE  ROOTS  OF  NUMBERS  FROM  0  TO  10,  AT  INTERVALS  OF  .1. 


N 

•••••i 

0 

.0 

^••^•M 

0 

.1 

.3i6 

.2 

.447 

.3 

.548 

.4 

.632 

.5 

.707 

.6 

"775 

.7 

~ 

.8 

.894 

.9 

.949 

i 

I.OOO 

i.  049 

1.095 

i.i4o 

i.i83 

1.225 

1.265 

i.3o4 

1.342 

1.378 

2 

i.4i4 

1.449 

1.483 

i.5i7 

1.549 

i.58i 

1.612 

1.643 

1.673 

1.703 

3 

1.732 

1.761 

1.789 

1.817 

1.844 

1.871 

1.897 

1.924 

1.949 

i.975 

4 

2.000 

2.025 

2.049 

2.074 

2.098 

2.  121 

2.i45 

2.168 

2.191 

2.214 

5 

2.236 

2.258 

2.280 

2.302 

2.324 

2.345 

2.366 

2.387 

2.408 

2.429 

6 

2.449 

2.470 

2.490 

2.5lO 

2.53o 

2.55o 

2.569 

2.588 

2.608 

2.627 

7 

2.646 

2.665 

2.683 

2.702 

2.720 

2.739 

2.7^7 

2.775 

2.793 

2.811 

8 

2.828 

2.846 

2.864 

2.881 

2.898 

2.915 

2.933 

2.950 

2.966 

2.983 

9 

3.000 

3.017 

3.o33 

3.o5o 

3.o66 

3.082 

3.098 

3.n4 

3.i3o 

3.i46 

SQUARE  ROOTS  OF  INTEGERS  FROM  10  TO  100. 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

m^mmmm* 

IO 

3.162 

I^T" 

3.464 

3.6o6 

3.742 

3.873 

4.000 

4.123 

4.243 

4.359 

20 

4.472 

4.583 

4.690 

4.  -796 

4.899 

S.ooo 

5.099 

5.196 

5.292 

5.385 

3o 

5.477 

5.568 

5.657 

5.745 

5.83i 

5.916 

6.000 

6.o83 

6.i64 

6.245 

4o 

6.325 

6.4o3 

6.48i 

6.557 

6.633 

6.708 

6.782 

6.856 

6.928 

7.000 

5o 

7.071 

7-Ui 

7.21  1 

7.280 

7.348 

7-4i6 

7-483 

7.55o 

7.616 

7.681 

60 

7-746 

7.810 

7.874 

7-9^7 

8.000 

8.062 

8.124 

8.i85 

8.246 

8.3o7 

70 

8.367 

8.426 

8.485 

8.544 

8.602 

8.660 

8.718 

8.775 

8.832 

8.888 

80 

8.944 

9.000 

9.o55 

9.1  10 

9.i65 

9.220 

9.274 

9.327 

9.38i 

9-434 

90 

9-487 

9.539 

9.592 

9-644 

9.695 

9-747 

9.798 

9.849 

9.899 

9.950 

z59 


TABLE    IX. 


THE  HYPERBOLIC  AND  EXPONENTIAL  FUNCTIONS  OF 
NUMBERS  FROM  0  TO  2.5,  AT  INTERVALS  OF  .1. 


X 

cosh  a? 

sinha? 

tanh  a? 

e* 

e  * 

0 

i  .000 

o 

o 

i  .000 

I  .000 

.  i 

i  .oo5 

.100 

.100 

i.io5 

.905 

.2 

i  .020 

.201 

.197 

I  .221 

.819 

.3 

i.o45 

.3o5 

.291 

i.35o 

.74i 

.4 

1.081 

.4n 

.38o 

1  .492 

.670 

.5 

1.128 

.521 

.462 

1.649 

.607 

.6 

i.tSS 

.637 

.537 

1.822 

.549 

•  7 

1.255 

.759 

.6o4 

2.014 

•497 

.8 

i.337 

.888 

.664 

2.226 

.449 

•9 

1.433 

1.027 

.716 

2  .46o 

.407 

1.0 

i.543 

1.176 

.762 

2.718 

.368 

.  i 

i  .669 

1.336 

.801 

3.oo4 

.333 

.2 

1.811 

i  .509 

.834 

3.320 

.3oi 

.3 

1.971 

1.698 

.862 

3.669 

.273 

.4 

2.l5l 

i  .904 

.885 

4.o55 

.247 

.5 

2.352 

2.129 

.905 

4.482 

.223 

.6 

2.577 

2.376 

.922 

4.953 

,202 

•7 

2.828 

2.646 

.935 

5.474 

.183 

.8 

3.107 

2.942 

•947 

6.o5o 

.i65 

•9 

3.4i8 

3.268 

•  956 

6.686 

.i5o 

2.0 

3.762 

3.627 

.964 

7.389 

.i35 

2.  I 

4.i44 

4.022 

.970 

8.166 

.  122 

2.2 

4.568 

4.457 

.976 

9.025 

.III 

2.3 

5.o37 

4.937 

.980 

9.974 

.  IOO 

2.4 

5.557 

5.466 

.984 

II  .023 

.091 

2.5 

6.i32 

6.o5o 

.987 

12.182 

.082 

1 60 


TABLE    X 

CONSTANTS 

MEASURES    AND    WEIGHTS 
AND    OTHER   CONSTANTS 


,  .... 


"*  ''  *' 

.*  „  •„ 


••  /*'      •"•   «•"••     -  •  -  - 

,.      /x,         ,   .,.      :„..  xx.'     .,     ., 

"•    •" 


'  '  '  '  '•"  •'  .....  '  '• 


- 


( 
rw^rf   ^r&rW^r 

,-'•>•<          •••'     :<•• 


•  >.. .  •  • '. 

'       •','•••  '  '' 


•' 


<v     v 


B' 

Si 

w  P- 


rt       P- 


IT 

f 


, Elements 
plans  and 


ottBtry 


Y  9    194 


May'47Pf 


M3O6247 


THE  UNIVERSITY  OF  CALIFORNIA  LIBRARY 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 
BERKELEY 

Return  to  desk  from  which  borrowed. 
This  book  is  DUE  on  the  last  date  stamped  below. 


REC'D  LD 

APR    9  1961 


30cf 


LD 

SEP  2  0  1962 


LD  21-100m-ll,'49(B7146sl6)476 


5 


V 

^~ 


' 3  ^   ul  L^, 


rr    / 


k-l>,ks-c,.        ru-y*- frf  0* 

, 

Oa  ju~r  ^*. 


t  X 


A  f  /MA4^e 

;    e,^    ot°  U.    T  -,H*" 

*fjj^cti4     *     */* 
v^-    J^^XA-^>  j     Wt^ 

UA_ 


